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# How many notes are in a scale? – How To Sing Notes On Sheet Music A major is an octave (4 notes in a scale) and a minor is an octave (4 notes in a scale). So there are 1, 2, 3, 12(5)(13)(15(5)…) notes in the scale, which is what this calculator tells you. However, in musical terms, the major and minor scales are both 3 notes in a set of 3 notes. So you can see how it’s not quite right in the first example to say it’s 12 notes in a scale. The whole difference between major and minor scales is that the minor uses a scale whose length is divided into three equal parts, while the major uses a scale whose length is divided only into two equal parts. (The idea of division in a way of how to divide a thing doesn’t exist in the physical world, because there’s only one place the thing can be spread). Some examples: It’s 10 days to a year You take a car to work and are told that all the parts that make up the car – such as tires and motors, chassis, seats, engine – are made of six pieces each weighing 1 kg. If you use the metric system or the old way of measuring, you’ve done six parts, but if you divide them up by three like the scale on the guitar, then it’s the same as if you used four bits instead of three. So you’d take the car to work but only take the part that weighs 6kg. It’s 1 inch thick This was true at one time (and still is), but in the past decade it’s not so true any more. If we use the old way of measuring, 1mm = 0.003 millimetres, that means the thickness of 1mm can be about 0.03 millimetres or even less. Nowadays a measure is used for most things – for example, the thickness of a piece of paper is used as a measure of thickness, rather than the thickness of a part, and a single millimetre is the thickness of about 1mm. So the thickness of a paper can be 0.03mm. Similarly a 1mm thick part will be 0.01mm, so it’s the same thickness. But the part itself can be less or larger than 1mm. If you’re unsure, use the “measure with confidence toolbox” to compare different methods of measuring thickness – and please remember that a number of measurement methods bbc learn to sing, can i learn to sing at 25, how to sing a song in hindi, how to learn singing in hindi, can you teach yourself to play guitar

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Video of the Day In captivity, pet rabbits don’t really need to do anything to attract a mate. That work is mostly done by default when a pair cohabit, or by the breeder, who selects which pairs to mate according to genetics and desired breed characteristics. In the wild, however, it’s a different story. Wild rabbits must go to greater lengths to woo potential mates. Eastern Cottontail Rabbit The most common species of wild rabbit in North America is Sylvilagus floridanus, more commonly known as the Eastern cottontail, according to Pennsylvania State University. Cottontails are relatively small rabbits, typically weighing 2 to 3 pounds and about 15 to 18 inches in length. Their fur ranges in color from light brown to dark gray, and they hold their long ears erect. This rabbit’s short, white, fluffy tail gives this species its common name. Cottontail Breeding Season Breeding season for cottontails begins each year in February or March and continues until September. Gestation lasts about 30 days, allowing the rabbits ample time to produce four or five litters a season. Litter size ranges from three to eight baby rabbits or kits, averaging four or five. The Mating Dance Before a litter's conceived, adult rabbits go through an interesting mating ritual in order to attract and select a mate. A male and a female, also known as a buck and a doe, perform a sort of dance in which the buck chases the doe until she stops, faces the buck and boxes him with her front paws. This goes on until one of the pair leaps straight into the air. The second rabbit also leaps into the air, completing the ritual and signalling that mating can now take place. Female Estrous Cycle With many mammals, mating times are determined by the female’s estrous cycle or ovulation cycle, when she is considered to be in heat. Rabbit does are somewhat unusual in their ovulation: It doesn’t occur until after they have mated. For this reason, sexually mature female rabbits are considered to basically be permanently in estrus and ready to mate. - DebMark Rabbit Education Resource: Breeding Rabbits - Pennsylvania State University: Virtual Nature Trail - State University of New York College of Environmental Science and Forestry: Eastern Cottontail - New Hampshire Public Television: Nature Works: Eastern Cottontail - Animal Diversity Web: Sylvilagus Floridanus - FAO.org: The Rabbit - Husbandry, Health and Production: Chapter 3 Reproduction - Jupiterimages/Photos.com/Getty Images

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# Visualizing complex number powers Learn how powers of complex numbers behave when you look at their graphical effect on the complex plane. ## Connection between $i^2 = -1$ and where $i$ lives We began our study of complex numbers by inventing a number $i$ that satisfies $i^2 = -1$, and later visualized it by placing it outside the number line, one unit above $0$. With the visualizations offered in the last article, we can now see why that point in space is such a natural home for a number whose square is $-1$. You see, multiplication by $i$ gives a $90^\circ$ rotation about the origin: You can think about this either because $i$ has absolute value $1$ and angle $90^\circ$, or because this rotation is the only way to move the grid around (fixing $0$) which places $1$ on the spot where $i$ started off. So what happens if we multiply everything in the plane by $i$ twice? It is the same as a $180^\circ$ rotation about the origin, which is multiplication by $-1$. This of course makes sense, because multiplying by $i$ twice is the same as multiplying by $i^2$, which should be $-1$. It is interesting to think about how if we had tried to place $i$ somewhere else while still maintaining its characteristic quality that $i^2 = -1$, we could not have had such a clean visualization for complex multiplication. ## Powers of complex numbers Let's play around some more with repeatedly multiplying by some complex number. ### Example 1: $(1 + i\sqrt{3})^3$ Take the number $z = 1 + i\sqrt{3}$, which has absolute value $\sqrt{1^2 + (\sqrt{3})^2} = 2$, and angle $60^\circ$. What happens if we multiply everything on the plane by $z$ three times in a row? Everything is stretched by a factor of $2$ three times, and so is ultimately stretched by a factor of $2^3 = 8$. Likewise everything is rotated by $60^\circ$ three times in a row, so is ultimately rotated by $180^\circ$. Hence, at the end it's the same as multiplying by $-8$, so $(1 + i\sqrt{3})^3 = -8$. We can also see this using algebra as follows: ### Example 2: $(1 + i)^8$ Next, suppose we multiply everything on the plane by $(1 + i)$ eight successive times: Since the magnitude of $1 + i$ is $|1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}$, everything is stretched by a factor of $\sqrt{2}$ eight times, and hence is ultimately stretched by a factor of $(\sqrt{2})^8 = 2^4 = 16$. Since the angle of $(1 + i)$ is $45^\circ$, everything is ultimately rotated by $8 \cdot 45^\circ = 360^\circ$, so in total it's as if we didn't rotate at all. Therefore $(1 + i)^8 = 16$. Alternatively, the way to see this with algebra is \begin{aligned} &\phantom{=}(1 + i)^8 \\\\ &= \left(\sqrt{2}\cdot(\cos(45^\circ) + i \sin(45^\circ) \right)^8 \\ &= (\sqrt{2})^8 \cdot \left( \cos(\underbrace{45^\circ + \cdots + 45^\circ}_{\text{8 times}}) + i\sin(\underbrace{45^\circ + \cdots + 45^\circ}_{\text{8 times}}) \right) \\\\ &= 16 \left(\cos(360^\circ) + i\sin(360^\circ) \right) \\\\ &= 16 \end{aligned} ### Example 3: $z^5 = 1$ Now let's start asking the reverse question: Is there a number $z$ such that after multiplying everything in the plane by $z$ five successive times, things are back to where they started? In other words, can we solve the equation $z^5 = 1$? One simple answer is $z = 1$, but let's see if we can find any others. First off, the magnitude of such a number would have to be $1$, since if it were more than $1$, the plane would keep stretching, and if it were less than $1$, it would keep shrinking. Rotation is a different animal, though, since you can get back to where you started after repeating certain rotations. In particular, if you rotate $\dfrac{1}{5}$ of the way around, like this then doing this $5$ successive times will bring you back to where you started. The number which rotates the plane in this way is $\cos(72^\circ) + i\sin(72^\circ)$, since $\dfrac{360^\circ}{5} = 72^\circ$. There are also other solutions, such as rotating $\dfrac{2}{5}$ of the way around: or $\dfrac{1}{5}$ of the way around the other way: In fact, beautifully, the numbers which solve the equation form a perfect pentagon on the unit circle: ### Example 4: $z^6 = -27$ Looking at the equation $z^6 = -27$, it is asking us to find a complex number $z$ such that multiplying by this number $6$ successive times will stretch by a factor of $27$, and rotate by $180^\circ$, since the negative indicates $180^\circ$ rotation. Something which will stretch by a factor of $27$ after $6$ applications must have magnitude $\sqrt[6]{27} = \sqrt{3}$, and one way to rotate which gives $180^\circ$ after $6$ applications is to rotate by $\dfrac{180^\circ}{6} = 30^\circ$. Therefore one number that solves this equation $z^6 = -27$ is \begin{aligned} \sqrt{3}(\cos(30^\circ) + i\sin(30^\circ)) &= \sqrt{3}\left(\frac{\sqrt{3}}{2} + i \frac{1}{2} \right) \\ &= \frac{3}{2} + i \frac{\sqrt{3}}{2} \end{aligned} However, there are also other answers! In fact, those answers form a perfect hexagon on the circle with radius $\sqrt{3}$: Can you see why? ## Solving $z^n= w$ in general Let's generalize the last two examples. If you are given values $w$ and $n$, and asked to solve for $z$, as in the last example where $n=6$ and $w = -27$, you first find the polar representation of $w$: $w = r(\cos(\theta) + i\sin(\theta))$ This means the angle of $z$ must be $\dfrac{\theta}{n}$, and its magnitude must be $\sqrt[n]{r}$, since this way multiplying by $z$ a total of $n$ successive times will in effect rotate by $\theta$ and scale by $r$, just as $w$ does, so $z = \sqrt[n]{r} \cdot \left( \cos\left(\dfrac{\theta}{n}\right) + i\sin\left(\dfrac{\theta}{n}\right) \right)$ To find the other solutions, we keep in mind that the angle $\theta$ could have been thought of as $\theta + 2\pi$, or $\theta + 4\pi$, or $\theta + 2k\pi$ for any integer $k$, since these are all really the same angle. The reason this matters is because it can affect the value of $\dfrac{\theta}{n}$ if we replace $\theta$ with $\theta + 2\pi k$ before dividing. Hence all the answers will be of the form $z = \sqrt[n]{r} \cdot \left( \cos\left(\dfrac{\theta + 2k\pi}{n}\right) + i\sin\left(\dfrac{\theta + 2k\pi}{n}\right) \right)$ for some integer value of $k$. These values will be different as $k$ ranges from $0$ to $n-1$, but once $k=n$ we can note that the angle $\dfrac{\theta + 2n\pi}{n} = \dfrac{\theta}{n} + 2\pi$ is really the same as $\dfrac{\theta}{n}$, since they differ by one full rotation. Therefore one sees all the answers just by considering values of $k$ ranging from $0$ to $n-1$.

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1 You visited us 1 times! Enjoying our articles? Unlock Full Access! Question # If the roots of the equation (c2−ab)x2−2(a2−bc)x+(b2−ac)=0 are real ane equal, show that either a=0 or (a3+b3+c3)=3abc. Open in App Solution ## Given, equation is: (c2–ab)x2–2(a2–bc)x+(b2–ac)=0To prove:a=0 or a3+b3+c3=3abc Proof: From the given equation, we have A=(c2–ab)B=–2(a2–bc)C=(b2–ac) It is being given that the equation has real and equal roots ∴D=0⇒B2–4AC=0 On substituting respective values of a, b and c in the above equation, we get [–2(a2–bc)]2–4(c2–ab)(b2–ac)=04(a2–bc)2–4(c2b2–ac3–ab3+a2bc)=04(a4+b2c2–2a2bc)–4(c2b2–ac3–ab3+a2bc)=0⇒a4+b2c2–2a2bc–b2c2+ac3+ab3–a2bc=0⇒a4+ab3+ac3–3a2bc=0⇒a[a3+b3+c3–3abc]=0⇒a=0 or a3+b3+c3=3abc Suggest Corrections 4 Join BYJU'S Learning Program Related Videos Solving QE by Factorisation MATHEMATICS Watch in App Join BYJU'S Learning Program

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# How do you find the value of the discriminant and determine the nature of the roots x^2 + 4x = -7? Sep 26, 2016 Roots are complex conjugates. For given equation they are $x = - 2 - 3 i$ or $x = - 2 + 3 i$ #### Explanation: The discriminant of a quadratic equation $a {x}^{2} + b x + c = 0$ is ${b}^{2} - 4 a c$, which decides the nature of roots of the equation. If $a$, $b$ and $c$ are rational and ${b}^{2} - 4 a c$ is square of a rational number, roots are rational. If ${b}^{2} - 4 a c > 0$ but is not a square of a rational number, roots are real but not rational. If ${b}^{2} - 4 a c > 0 - 0$ we have equal roots. If ${b}^{2} - 4 a c < 0$ roots are complex, and if $a$, $b$ and $c$ are rational. they are complex conjugates In ${x}^{2} + 4 x = - 7 \Leftrightarrow {x}^{2} + 4 x + 7 = 0$ the discriminant is ${4}^{2} - 4 \times 1 \times 7 = 16 - 28 = - 12$ hence roots are complex conjugates. In fact ${x}^{2} + 4 x + 7 = 0$ $\Leftrightarrow {x}^{2} + 4 x + 4 - \left(- 3\right) = 0$ or ${\left(x + 2\right)}^{2} - \left(3 {i}^{2}\right) = 0$ or $\left(x + 2 + 3 i\right) \left(x + 2 - 3 i\right) = 0$ i.e. $x = - 2 - 3 i$ or $x = - 2 + 3 i$

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Scientists from the University of California at Santa Cruz and the Carnegie Institution for Science in the U.S have discovered a super-Earth that may have extraterrestrial life. Scientists have discovered a potentially habitable super-Earth along with three other planets orbiting a nearby star and it is just 22 light years away from Earth. Super-Earth takes about 28 days to orbit its Sun-like star and its minimum mass is 4.5 times that of Earth. They found that the planet temperature is neither too hot nor too cold for liquid water to exist on the planet's surface and where there is water they will surely be life. Super Earth orbits around Sun-Like star, GJ 667C which is an M-class dwarf star and the host star is a member of a triple-star system, is where a planet revolves that around three different stars. The other two stars, GJ 667A and GJ667B in the triple-star system are a pair of orange K dwarfs, that contains concentration of heavy elements such as iron, carbon, and silicon which is 25 per cent compared to our Sun. Such elements are the building blocks for rocky planets. According to the scientists, super-Earth receives only 90 percent of the light compared to Earth. However, the incoming light is in the infrared, a higher percentage of this incoming energy should be absorbed by the planet. When both these effects are taken into account, the planet is expected to absorb about the same amount of energy from its star that the Earth absorbs from the sun. Scientists have also found a system that might contain a gas-giant planet and an additional super-Earth with an orbital period of 75 days. However, further observations are needed to confirm these two possibilities. "This planet is the new best candidate to support liquid water and, perhaps, life as we know it," said Anglada-Escudé from the Carnegie Institution for Science. "With the advent of a new generation of instruments, researchers will be able to survey many M dwarf stars for similar planets and eventually look for spectroscopic signatures of life in one of these worlds," he said. "This was expected to be a rather unlikely star to host planets. Yet there they are, around a very nearby, metal-poor example of the most common type of star in our galaxy," said Steven Vogt, a professor of astronomy and astrophysics at Univerisity of California. "The detection of this planet, this nearby and this soon, implies that our galaxy must be teeming with billions of potentially habitable rocky planets."

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Harmonic Intervals on the Piano Staff Until now we played separated notes written on the musical staff. The intervals between these notes create a certain MELODY. That's why they're called MELODIC INTERVALS. A melody is a tune basically. It's a succession of melodic intervals arranged in a certain musical shape that makes sense to us. The melody is the most dominant element in a composition. Until today we used to play the melody with both hands while moving from the upper piano staff to the lower piano staff. In today's piano lesson the melody will be played with the right hand, written on the upper piano staff. Locating the melody in the higher register of the piano makes it sound more expressive. What are we going to do with the left hand then? We'll play the harmony! When notes are played together they make HARMONY. As opposed to melodic intervals played separately, HARMONIC INTERVALS are played together. The harmony is a type of accompaniment. This combination of the harmonic intervals with the melody, produces chords (You must have at least three notes in a chord). The middle register is ideal for the harmony, and therefore, the left hand will be playing the harmony part at the moment, written on the lower piano staff. Before we go on there's another important element in music I'd like us to discuss - the BASS. The bass is the lowest part in a musical composition. The bass tells us the name of the chord and later on we'll learn that you can tell how a chord functions in a chord progression according to the bass note of the chord. Being the lowest part in a composition, the bass naturally goes to the lowest piano staff so it is played with the left hand. Later on, in our piano course we'll learn about scales, and from that point on we'll learn to realize the logic behind different scale degrees. Now, you might wonder how it is possible to play the melody, the harmony and the bass when all we have is two hands. How to combine all three elements? Well, think of it... If you play two notes at the same time, a melody note in the right hand and a bass note in the left hand you're already creating harmony (It only takes two notes to create an harmonic interval). let's say that instead of playing one note in the left hand we'll be playing the bass and an extra note. We then have an harmonic interval of two notes in the left and the melody in the right hand. You get three elements played with two hands. The example above of the song Merry had a little lamb shows how it's done nicely. How to Practice Notes Reading When Both Piano Staffs Are Notated at the Same Time? When we try to read piano notes on both the treble piano staff and the bass staff there's a lot of information we need to process at the one time. The best way to practice reading notes on the piano staff is to FIRST READ AND PLAY EACH HAND APPART. It's just like a relationship in real life really... In a partnership each side of the relationship has to know his own part before he can communicate and connect with the other. Therefore we will first learn to read and play with each hand apart until we play the parts written on each piano staff fluently and only then we'll try to read both the treble and bass clef together. Before we go on pay attention to a new hand position that will help us to play pieces with harmonic interval on the grand staff. We the right thumb on middle C and the left thumb on the lower C. Here below you'll find some great beginner piano sheet music with harmonic intervals on the staff. I recommend them highly. They're so fun! Beginner Pieces for the Grand Staff with Harmonic Intervals. |Name of the Piece||Audio File||Video File| |Away in the Deep Forest||Download||Play| At this point many students start to encounter difficulties in combining between the right hand and the left hand. The problem is that you have to play these pieces smoothly while keeping the independence and coordination of your fingers while gaining each times a faster finger speed. To learn to this you must check out the Hannon Finger Exercises Piano Course. In this DVD, the exercises are demonstrated by an accomplished jazz instructor, James Wrubel, and broken down step-by-step in VIDEO format so that anyone with a dvd player or computer and 1 hour to spare can start using them right away. I highly recommend you check out a sample of the first exercise on their website. Just that one sample exercise can dramatically improve your speed and precision. Click here to check out the Hannon Finger Exercises Piano Course Now!

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In cryptography, a private or secret key is an encryption/decryption key known only to the party or parties that exchange secret messages. In traditional secret key cryptography, a key would be shared by the communicators so that each could encrypt and decrypt messages. The risk in this system is that if either party loses the key or it is stolen, the system is broken. A more recent alternative is to use a combination of public and private keys. In this system, a public key is used together with a private key. See public key infrastructure (PKI) for more information. By submitting your email address, you agree to receive emails regarding relevant topic offers from TechTarget and its partners. You can withdraw your consent at any time. Contact TechTarget at 275 Grove Street, Newton, MA. Continue Reading About private key (secret key) - Marc Branchaud's thesis, A Survey of Public Key Infrastructures , includes a tutorial on how public key cryptography works and compares several PKI approaches. - IBM's Introduction to Cryptography also mentions private keys as part of a public key infrastructure.

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SNAP, the Supplemental Nutrition Assistance Program, is changing. In December, the current administration announced a new rule that would limit States’ ability to set eligibility requirements, instead applying federal work requirements to qualify for SNAP benefits. The first attempt at a Food Stamp program began back in 1939. It was designed to allow citizens to buy “surplus” foods at a reduced rate based on how much full-rate groceries they purchased. Though the 1939 plan didn’t last, President Kennedy was able to revive a version of it in 1961. That version ultimately became the Food Stamp Act of 1964. It was designed to strengthen agriculture, provide improved nutrition among low-income households, and allow states to set their own eligibility standards. SNAP (as the Food Stamp Act came to be known) became a means of ensuring that no American ever had to be severely hungry or undernourished. According to feedingamerica.org, “SNAP, in conjunction with food banks, community groups, churches and volunteer organizations, collectively helps strengthen communities by providing the fuel and nutrition people need.” SNAP supports nearly 10 million families nationwide, but there are some misconceptions about it and how it functions. 4 out of 5 SNAP recipients have jobs Four out of Five SNAP recipients work low-wage jobs or jobs that have inconsistent availability. When data on SNAP participants’ is looked at, it shows workers earning low wages are frequently in and out of work and so are on and off SNAP as their earnings fall and rise. Most low-income, non-disabled adults work, often with interruptions, and so are more likely to participate in SNAP when they are not working. For the small share of participants who are unable to work, SNAP is a vital way for them to buy groceries. Administering SNAP meets high ethical standards The earliest form of the food stamp program was plagued with corruption and mismanagement. Current SNAP operates very efficiently with 92% of all funding going directly to food for qualified beneficiaries. SNAP helps farmers The USDA has shown that SNAP also helps keep up demand for farm products and food, thereby boosting growth and jobs. Food Stamps can’t be used to buy non-nutritional items A handful of states operate a SNAP Restaurant Meal Program, but they only allow elderly, homeless, and disabled persons to purchase ready-to-eat foods. SNAP benefits can be used to buy baby formula, but NOT other household items (soaps, paper products, pet foods, alcohol or nicotine products). The focus is on nutritional support. Fewer people are relying on SNAP already SNAP caseloads have been falling by bigger and bigger degrees since 2013. Since then, more than 7 million people have voluntarily come off of the program. This is evidence that SNAP works as a means of getting people through difficult times and encouraging self-reliance. By the administration’s own estimates, their new ruling will make 700,000 more people disqualify for SNAP before they are ready. This is why fourteen states, New York City and the District of Columbia have sued the Trump administration to block the new rules from going into effect this April. Catholic Charities feeds the hungry Food, shelter, clothing and fuel are the four essentials to life Henry David Thoreau identified. Catholic Charities of New York has always provided food support to hungry New Yorkers and always will, regardless of what other support is or is not available publicly. CCNY operates food banks from New York City through the Hudson Valley. It keeps food pantries in operation and collaborates with agencies like the NYC Coalition Against Hunger. Catholic Charities’ Feeding Our Neighbors program helps to support local pantries, like that operated at the St. Augustine location in the Bronx. Click on our google map below to find a location near you. No one can ever be independent and self-reliant if they don’t know where their next meal is coming from.

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# Simplifying Fractions using "Keep,Change,Flip" In the words of Flocabulary, " Keep, change, Flip, yea that's the action, everybody's gonna know how were dividing fractions" In this video you will learn how to simplify complex fractions. A complex fraction sounds difficult, but it is simply a fraction on top of a  fraction. I call them a double decker fraction. "Keep, Change, Flip" is a technique used when dividing fractions. You keep the first fraction, change the division sign to multiplication, and flip the second fraction. The video solves three problems. Problem 1. 5+7/11 /5/12 In this problem you can use " Keep, Change,Flip" in order to solve. Keep the first fraction,change the division sign to multiplication, and create a reciprocal by flipping the last fraction. Problem 2. 3/4 + 2/3 /1/2 In this problem I show how to simplify the top fraction, rewrite the fraction, and again apply the "Keep,Change, Flip" technique. Problem 3. Involves a complex fraction with a variable. Again, "Keep,Change,Flip" can be used to solve this complex fraction. Enjoy this video by Flocabulary for a review of " Keep,Change,Flip" Follow moomoomath's board Math Resources and helpful information on Pinterest. #### Post a Comment Powered by Blogger. Back to Top

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General Information on Mold and Dampness There is always some mold everywhere - in the air and on many surfaces. Molds have been on the Earth for millions of years. Mold grows where there is moisture. Exposure to damp and moldy environments may cause a variety of health effects, or none at all. Some people are sensitive to molds. For these people, molds can cause nasal stuffiness, throat irritation, coughing or wheezing, eye irritation, or, in some cases, skin irritation. People with mold allergies may have more severe reactions. Immune-compromised people and people with chronic lung illnesses, such as obstructive lung disease, may get serious infections in their lungs when they are exposed to mold. These people should stay away from areas that are likely to have mold, such as compost piles, cut grass, and wooded areas.

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Physical activity and nutrition play a crucial role in overall health and wellbeing. However, their impact on resilience, the ability to adapt and bounce back from difficult experiences, is still not fully understood. A recent cross-sectional study conducted in the Netherlands aimed to explore the relationship between physical activity, nutrition, and resilience. The findings of this study have significant implications for promoting resilience and overall mental health in the population. Resilience is a dynamic process that involves the ability to maintain or regain mental health and wellbeing in the face of adversity. It has been linked to various positive outcomes, including lower levels of anxiety and depression, better coping skills, and improved overall quality of life. Given the potential benefits of resilience, understanding the factors that influence it is of great importance. Physical activity has long been recognized as an important factor in promoting physical health. It has been associated with a range of benefits, including reduced risk of chronic diseases such as cardiovascular disease and diabetes, improved cardiovascular fitness, and enhanced mental wellbeing. In recent years, research has also begun to uncover the role of physical activity in promoting resilience. The cross-sectional study in the Netherlands surveyed a sample of adults to measure their levels of physical activity and resilience. The participants were asked to report on their weekly physical activity levels, including both moderate-intensity activities such as brisk walking and vigorous-intensity activities such as running or swimming. Additionally, they were assessed using a standardized questionnaire to measure their resilience levels. The results of the study revealed a significant association between physical activity and resilience. Participants who engaged in higher levels of physical activity were found to have higher resilience scores compared to those who were less active. This finding underscores the potential impact of physical activity on mental wellbeing and the ability to adapt to stress and adversity. The study also examined the role of nutrition in relation to resilience. Nutrition plays a critical role in supporting overall health and wellbeing, and its impact on mental health has been well-documented. However, its specific relationship to resilience has been less studied. The findings of the study revealed a positive association between healthy dietary patterns and resilience. Participants who reported consuming a diet rich in fruits, vegetables, whole grains, lean proteins, and healthy fats had higher resilience scores compared to those with less healthy dietary habits. These findings highlight the importance of nutrition in supporting mental wellbeing and resilience. A well-balanced diet that provides essential nutrients and supports overall health can contribute to better mental health outcomes and the ability to cope with stress and adversity. The researchers also noted that the combined impact of physical activity and nutrition on resilience may be greater than the sum of their individual effects, suggesting that a healthy lifestyle that includes both regular physical activity and a nutritious diet may have the most significant impact on resilience. The implications of these findings are significant, particularly in the context of promoting mental health and resilience in the population. By understanding the role of physical activity and nutrition in supporting resilience, public health efforts can be better tailored to address these factors. Promoting regular physical activity and healthy dietary patterns can be integrated into public health initiatives aimed at enhancing resilience and overall mental wellbeing. In addition to promoting individual behaviors, the findings of the study also have implications for larger societal and environmental factors that impact physical activity and nutrition. Access to safe and accessible spaces for physical activity, as well as policies and interventions that support healthy eating habits, can contribute to improving resilience at the population level. These broader efforts to create environments that support healthy behaviors can have a positive impact on mental health and resilience across communities. It is important to note that the cross-sectional nature of the study means that it cannot establish causality. While the findings indicate an association between physical activity, nutrition, and resilience, further research is needed to understand the specific mechanisms underlying this relationship. Longitudinal studies that follow individuals over time can provide more insight into the long-term impact of physical activity and nutrition on resilience. Nevertheless, the findings of this study provide valuable insights into the potential impact of physical activity and nutrition on resilience. By promoting healthy lifestyle behaviors, individuals and communities can work towards building greater resilience and enhancing mental wellbeing. This research adds to the growing body of evidence supporting the importance of healthy behaviors in promoting mental health and resilience, and underscores the potential for targeted interventions to improve outcomes in this area.

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Low-stakes assignments are ungraded or carry little weight in grading. For example, you might average grades for all low-stakes assignments into one homework grade; or you might grade using a simple P/F scale. Low-stakes assignments fulfill four goals: - They provide students with ample practice and fluency in preparation for higher-stakes assignments. Often, low-stakes work serves as a form of prewriting, that is, a method for developing a topic or for thinking up a topic. - They help students engage in and learn the course content. Some studies show students feel more deeply engaged in their learning when they are required to write or speak to others about it. - They give students the opportunity to reflect, both on the course content and on the composing process itself. Reflection aids in thinking through a topic or an argument and leads to growth in critical thinking skills and, in turn, writing or speaking skills. - They can soothe the anxious student or motivate the reluctant student. Although low-stakes assignments are often not graded, or graded minimally, they are most meaningful if accompanied by feedback. The reader/responder does not have to be the instructor; graders, peers, or writing center consultants can be helpful, or students can self-assess. Types of Assignments Any assignment can be low stakes—an ungraded rough draft or speech outline, for example. However, some assignments are more commonly used in providing practice and helping students achieve fluency. These include field notes, reading logs, journals , discussion list responses, two-minutes papers or speeches, and mini essays Many techniques used to develop content for papers or speeches can also be used for low-stakes speaking and writing assignments. For example, freewriting, brainstorming, and answering heuristic questions (Who? What? When? Where? Why?) can be the basis for effective writing and small group discussion activities. Keep in mind that low-states assignments should specify a rhetorical situation (audience, purpose, genre), just as any other assignment would. However, since the purpose of low-stakes writing often is to promote writing-to-learn, the audience and purpose sometimes are simply to display knowledge. Still, you might want to clarify, when you are the specified audience, the assignment's learning outcomes. If you want students to show critical thinking in their journals, explain that as a reader, you will be looking for examples of critical thinking. Showing them what you mean by critical thinking through concrete examples will help even more. Likewise, if you want them to demonstrate they retained the gist of a reading, explain that is what you will look for when you review their reading log. Responding to Low-Stakes Assignments No matter who responds, the most useful feedback is (1) specific and detailed; (2) honest; and (3) tactful. Responses to low-stakes assignments can come in a number of forms, including: (1) marginal comments or questions that respond to content or evidence; (2) check marks next to errors that interfere with comprehension; and (3) rubrics designed to guide reading and response. It is a good idea to make a distinction for students between how you respond to low-stakes assignments and how you grade high-stakes assignments. Students need to understand that ungraded responses are not comprehensive reviews of everything that needs to be done to improve a draft or a performance. They hold ultimate responsibility for changes. Some things a reviewer (including an instructor) suggests might be inadequate or even wrong, or one change they make because of a reviewer's suggestion might make further changes necessary. Never center responses on errors in the early stages of the composing process. If students are to feel that revision should be deep and meaningful, they should concentrate on ideas, content, and argument first. Rather than point out specific errors in logic or punctuation, the responder can let the student know he/she needs to concentrate on those areas. Pointing out one or two errors as examples should be sufficient. "Low-Stakes Writing and Critical Thinking

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Ganguli | Math 1375 | Fall 2020 If you would like to review the material that’s on Quiz #3, I recommend that you study Example 10.7 in the textbook–especially 10.7(a): In this example, you are presented with various polynomials and asked to find the roots, and use them to factor the polynomial completely. Let’s take a closer look at the polynomial in 10.7(a): the cubic polynomial f(x) = 2x3 – 8x2 – 6x + 36. How do we find the roots of this cubic? As you can see in the textbook explanation, we can start by looking at the graph! Here’s a nicer version of the graph I created in Desmos: Note that I factored the common factor of 2 out of the polynomial. That makes the algebra a little bit simpler going forward… Now, as the textbook explains as well, from looking at the graph it seems like x = -2 and x = 3 are roots. But to be sure we should check algebraically, i.e., by evaluating f(-2) and f(3), as they do in the textbook. The algebra is (just a bit) simpler with the 2 factored out: f(3) = 2*( 3^3 – 4*(3^2) – 3*3 + 18) = 2*(27 – 36 – 9 + 18) = 2*0 = 0 Now how can we use the roots to factor the polynomial? That’s where the “Factor Theorem” comes in. It’s stated in Sec 8.2 of the textbook (read that section!); here is a statement via wikipedia: “The factor theorem states that a polynomial  has a factor  if and only if   (i.e.   is a root).” Note that k here represents a constant value for the input variable x. So in our example, since we know that k = 3 is a root of f(x), therefore we know that (x – 3) is a factor of f(x)! Similarly, since k = -2 is a root of f(x), we know that (x – (-2)) = (x + 2) is a factor of f(x). How can we use that information to actually factor f(x)? By long division! In this case, we would set up long division in order to compute either f(x) ÷ (x – 3) or f(x) ÷ (x + 2) In the textbook (see the bottom of p136) they carry out the long division f(x) ÷ (x-3) to show that f(x) = (x – 3)(2x2 – 2x – 12) Here’s the long division for (x3 – 4x2 – 3x + 18) ÷ (x + 2) (I’m leaving out the factor of 2 from f(x) for the long division, but then put it back in at the bottom when factoring f(x)): Therefore, we conclude that f(x) = 2(x + 2)(x2 – 6x + 9) and in this case we can factor the quadratic to get: f(x) = 2(x + 2)(x2 – 6x + 9) = 2(x + 2)(x – 3)(x – 3) = 2(x + 2)(x – 3))2 This shows that the only roots of f(x) are x = -2 and x = 3 (where the latter is a root of multiplicity 2), and thus (as the Desmos graph seemed to show, but which we have now proved algebraically): the only x-intercepts of the graph are at (-2, 0) and (3,0). Also note that we can easily find the y-intercept of the graph by computing f(0): f(0) = 2*( 0^3 – 4*(0^2) – 0*3 + 18) = 2*(18) = 36 i.e., the y-intercept is at (0, 36), again as indicated by the Desmos graph. ## 1 Comment 1. Suman Ganguli Note: I just updated this post with the polynomial long division, and showed how that allows us to get to the fully factored form (and hence all the roots) of the given polynomial. Let me know if you have any questions! Also a reminder that the “Rational Functions – Domains” WebWork set is due tonight, so finish that first if you haven’t already, and then complete the quiz (due Sunday). ### 1 Pingback Theme by Anders NorenUp ↑

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Free online geometry homework help Free online geometry homework help is a software program that supports students solve math problems. Keep reading to learn more! The Best Free online geometry homework help Math can be a challenging subject for many students. But there is help available in the form of Free online geometry homework help. Solving log equations is a common problem in which the relationship of the logarithm and base is not clear. When solving log equations, remember that you can use basic logic to determine whether or not the equation is correct. When you have an unknown log value, simply subtract the value from 1 and then divide by the base. If your answer is positive, then your equation is correct. If your answer is negative, then your equation is incorrect. For example: Consider the following equation: If we want to solve it, we can see the two values are 100 and -2. Then: Now if we take out 100 (because 2 0), and divide by base 2 (because -1 0): Now we know that it’s incorrect because it’s negative, so we can solve it with a log table as follows: As you can see, all values are negative except 1. So our solution is as follows: We get 0.0132 0 0.0421 1, so our solution for this equation is correct. Solving problems is something that's a part of being human. We all need to solve problems in our lives; whether they be problems at work, at home, or with our relationships. And when you're able to solve problems, it can make you feel good about yourself and can even help you achieve other goals. There are lots of different ways to solve problems. You can talk to someone about your problem, try to find a solution on your own, or do both. If you want to be really good at solving problems, it's important to learn how to listen and ask questions, as well as how to use your imagination and think outside the box. And when you know how to solve problems well, you'll be able to get more done in less time. Other people solve problems by identifying the source of the problem. For example, if you want to commute to work on time, you can find out how long it takes to commute on public transportation and then try to figure out how you can cut down that commute time. In general, solving means finding a way to get something done. There are different types of solving: analytical solving, creative solving, critical thinking solving, etc. Analytical solving is when you use your logic and thinking skills to solve problems. Creative solving is when you use your creativity and imagination to solve problems. Critical thinking solving is when you use your critical thinking skills to solve problems. There are many other ways that people solve problems as well, but these four are some of the most common ways. There are a few different methods that can be used to solve multi step equations. The most common method is to use the distributive property to simplify the equation and then solve it by using the order of operations. Another method is to use inverse operations to solve the equation.

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Home » What Fraction Is Equivalent To -3/2? # What Fraction Is Equivalent To -3/2? ### Which is equivalent to fractions? Fractions represent a part of a whole. Equivalent fractions represent the same portion of the whole, even though they look different. To find an equivalent fraction, you simply multiply both the numerator and denominator by the same number. Think of it like this: imagine you have a pizza cut into 8 slices. You eat 2 slices, which is the same as eating 2/8 of the pizza. Now, imagine someone else cuts the same pizza into 16 slices. If they eat 4 slices, that’s the same as 4/16 of the pizza. Even though they ate different numbers of slices, they both ate the same amount of pizza. 2/8 and 4/16 are equivalent fractions. You can create an infinite number of equivalent fractions for any given fraction by multiplying both the numerator and denominator by the same number. This doesn’t change the value of the fraction, it just changes the way it looks. Here are some examples: * 1/2 is equivalent to 2/4, 3/6, 4/8, and so on. * 3/4 is equivalent to 6/8, 9/12, 12/16, and so on. To understand if two fractions are equivalent, you can simplify them. Simplifying a fraction means dividing both the numerator and denominator by their greatest common factor (GCF). If the simplified fractions are the same, then the original fractions are equivalent. For example, to simplify 6/8, you can divide both the numerator and denominator by 2, which gives you 3/4. Since 3/4 is the simplest form of both 6/8 and 3/4, they are equivalent fractions. Understanding equivalent fractions is important because it helps you to work with fractions more easily. For example, when adding or subtracting fractions, it’s often helpful to find equivalent fractions with a common denominator. ### What fraction is equivalent to 3800 + 19000? Let’s break down how to find the fraction equivalent to 3800 + 19000. First, we need to simplify the expression. 3800 + 19000 = 22800. Now, we’re looking for a fraction that represents 22800. To do this, we’ll find the greatest common factor (GCF) of both 22800 and 1. The GCF is the largest number that divides evenly into both numbers. In this case, the GCF is 1. Now, we divide both the numerator and denominator by the GCF: 22800 ÷ 1 = 22800 and 1 ÷ 1 = 1. Therefore, the fraction equivalent to 3800 + 19000 is 22800/1. Fractions represent a part of a whole. The top number (numerator) tells you how many parts you have, and the bottom number (denominator) tells you how many parts make up the whole. In this case, the fraction 22800/1 indicates that we have 22800 parts out of a whole that consists of only 1 part. Essentially, this means we have 22800 times the whole. Fractions can be simplified by dividing both the numerator and denominator by their greatest common factor. This simplifies the fraction without changing its value. It makes the fraction easier to understand and work with. ### What is 1 ⁄ 3 called? You’re probably wondering what one third is called, right? It’s a simple fraction that represents one out of three equal parts of something. Think of cutting a pizza into three slices – one third represents one of those slices. In math, we can express one third in a few ways: 1/3 is the most common way to write it as a fraction. 0.333333333… is its decimal representation, which goes on forever. So, next time you see one third, you’ll know it’s just another way to say one out of three equal parts. It’s important to note that one third is a rational number because it can be expressed as a fraction. It’s also a repeating decimal because its decimal form goes on forever and repeats the digit 3. Here’s a fun fact: one third is one of the simplest fractions, but it can also be tricky to work with because it results in a repeating decimal. This makes it a bit different from other fractions like one half (1/2), which has a simple decimal representation (0.5). Now that you know what one third is called, you can confidently use it in your everyday conversations and calculations! ### How many 3/4 are in 1? We’re trying to figure out how many times 3/4 fits into 1. Think of it like cutting a pie into four slices. 3/4 represents three of those slices, and we want to know how many times we can fit three slices into the whole pie. You can figure this out by dividing 1 by 3/4. When dividing by a fraction, you flip the fraction and multiply. This means we’re actually multiplying 1 by 4/3. 1 * 4/3 = 4/3 This is an improper fraction, meaning the numerator (top number) is larger than the denominator (bottom number). We can convert it to a mixed number by dividing the numerator by the denominator. 4 / 3 = 1 with a remainder of 1 This means that 4/3 is equal to 1 1/3. So, there are 1 1/3 three-quarters in one. Let’s break down why this works. Imagine you have a pizza cut into four slices. You want to figure out how many groups of three slices you can make. You can make one full group of three slices (3/4 of the pizza). You’ll have one slice left over, which is 1/4 of the pizza. That leftover slice is one-third of a full group of three slices, This is why the answer is 1 1/3. We can also think about this in terms of decimal numbers. 3/4 is equal to 0.75, and 1 is equal to 1.00. We can then divide 1.00 by 0.75 to get 1.3333…, which is the decimal equivalent of 1 1/3. ### What is 3/8 called? Three eighths is the name for the fraction 3/8. It can also be written as 0.375 in decimal form. This fraction represents three out of eight equal parts of a whole. Think of it like slicing a pizza into eight equal pieces. Three eighths means you have three of those slices. Fractions are a way to express parts of a whole. They are written as one number (the numerator) over another number (the denominator). The numerator tells you how many parts you have, and the denominator tells you how many parts the whole is divided into. You can use fractions to represent many different things, like: Parts of a whole: For example, 3/8 of a pizza. Ratios: For example, a ratio of 3 boys to 8 girls in a classroom. Division: For example, 3 divided by 8. In everyday life, you might encounter three eighths in situations like: Measuring ingredients: A recipe might call for three eighths of a cup of flour. Time:Three eighths of an hour is 22.5 minutes. Money:Three eighths of a dollar is 37.5 cents. Understanding fractions is a valuable skill that can help you solve many everyday problems. ### Is 0.19 as a fraction? You’re in luck! Converting decimals to fractions is pretty straightforward. 0.19 as a fraction is 19/100. Let’s break it down: Decimal Places: The decimal 0.19 has two digits after the decimal point. This tells us the denominator of our fraction will be 100 (since 100 has two zeros). Numerator: The digits after the decimal point, 19, become the numerator of our fraction. So, we get 19/100. Since 19 and 100 share no common factors other than 1, this fraction is already in its simplest form. Think of it like this: 0.19 means “nineteen hundredths,” and that’s exactly what the fraction 19/100 represents. ### What is 10.3% as a fraction? Let’s figure out how to write 10.3% as a fraction. First, remember that a percentage means “out of one hundred.” So, 10.3% is the same as 10.3 out of 100. To write this as a fraction, we put 10.3 over 100: 10.3/100. But fractions are usually simplified. To do that, we need to get rid of the decimal in the numerator. Since 10.3 has one decimal place, we can multiply both the numerator and denominator by 10 to move the decimal one place to the right: (10.3 * 10) / (100 * 10) = 103/1000 Now, we see if there’s a common factor between 103 and 1000 that we can divide by to simplify further. In this case, there isn’t. So, the simplest form of 10.3% as a fraction is 103/1000. Understanding Percentages and Fractions It’s important to understand the relationship between percentages and fractions. A percentage represents a part of a whole, just like a fraction. Think of a pizza cut into 100 slices. If you eat 10 slices, you’ve eaten 10% of the pizza. Similarly, eating 10.3 slices represents 10.3% of the pizza. Fractions offer a more precise way to express parts of a whole compared to percentages. For example, it’s easier to visualize 1/4 of a pie than 25% of a pie. When converting percentages to fractions, remember that the percentage represents the numerator (the part), and 100 always represents the denominator (the whole). Understanding how to convert between percentages and fractions is helpful in various situations, including calculations involving proportions, discounts, interest rates, and much more. ### What is an example of an equivalent fraction? Equivalent fractions are fractions that represent the same value even though they have different numbers in the numerator and denominator. For example, 1/2 is equivalent to 4/8, even though they have different numbers. To find equivalent fractions, you can multiply or divide both the numerator and denominator of the original fraction by the same number. This is like slicing a pizza into different sizes – if you cut it in half, you get two equal pieces, but you can also cut it into four equal pieces. Both represent the same amount of pizza. Let’s look at how we can find equivalent fractions using this method. If we start with the fraction 1/2 and want to find an equivalent fraction with a denominator of 8, we can multiply both the numerator and denominator by 4. This gives us (1 x 4) / (2 x 4) = 4/8. You can also use this method to simplify fractions. For example, if you have the fraction 4/8, you can divide both the numerator and denominator by 4 to get (4 / 4) / (8 / 4) = 1/2. Understanding equivalent fractions is important in many areas of math, including simplifying fractions, comparing fractions, and solving equations. It’s also useful in everyday life, such as when you’re trying to figure out how much of something you need or how to divide something fairly. ### How do you find equivalent fractions? Finding equivalent fractions is like finding different ways to express the same amount. Think of a pizza! If you cut the pizza into 4 slices and take 2 slices, you’ve eaten 2/4 of the pizza. If you cut the same pizza into 8 slices and take 4 slices, you’ve still eaten the same amount: 4/8 of the pizza! Equivalent fractions are fractions that represent the same portion or value, even though they look different. The trick to finding them is to multiply or divide both the numerator and the denominator by the same number. For example, if you want to find an equivalent fraction for 2/4, you can multiply both the numerator and denominator by 2. This gives you 4/8, which is an equivalent fraction. Important Note: The key is that you must multiply or divide by the same number. Otherwise, you’ll end up with a different value. Let’s try another example: Let’s say we want to find an equivalent fraction for 6/9. To make the numbers smaller, we can divide both the numerator and denominator by 3. This gives us 2/3, an equivalent fraction. Here’s why this works: Multiplying the numerator and denominator by the same number is essentially multiplying the fraction by 1 (in the form of a fraction, like 2/2 or 3/3). Since multiplying by 1 doesn’t change the value, the resulting fraction is equivalent. The same logic applies to division. Dividing by the same number is like dividing by 1, which again doesn’t change the value. Remember, you can find infinitely many equivalent fractions for any given fraction! Just keep multiplying or dividing the numerator and denominator by the same whole number. ### Why are equivalent fractions the same? You’re right, equivalent fractions look different but have the same value! This is because they represent the same portion of a whole. Here’s why: Imagine a pizza cut into eight slices. You eat two slices. You’ve eaten 2/8 of the pizza. Now, let’s say we cut the pizza in half, creating four slices. You would still have eaten 2/8 of the pizza, but now you’ve also eaten 1/4 of the pizza. This is because 2/8 and 1/4 are equivalent fractions. Equivalent fractions are made by multiplying or dividing both the numerator and denominator by the same number. In the pizza example, we divided both the numerator and the denominator of 2/8 by two (2 ÷ 2 = 1 and 8 ÷ 2 = 4) to get 1/4. It’s like multiplying a number by 1. Multiplying by 1 doesn’t change the value, right? Dividing the numerator and denominator by the same number is essentially multiplying by a fraction that equals 1 (like 2/2 or 3/3). Let’s look at another example. We can make 3/6 equivalent to 1/2. To do this, we divide both the numerator and denominator by 3. 3 ÷ 3 = 1 and 6 ÷ 3 = 2. So, 3/6 and 1/2 are equivalent fractions. Understanding equivalent fractions is crucial for simplifying fractions and for comparing and adding fractions with different denominators. ### What are two fractions that are different but equivalent? Let’s explore the concept of equivalent fractions, which are fractions that represent the same value even though they look different. You can think of them as different ways of slicing up the same pie. Equivalent fractions are created by multiplying both the numerator and denominator of a fraction by the same number. This is like cutting your pie into more slices, but the size of each slice is smaller. For example, take the fraction 1/2. If we multiply both the numerator and denominator by 2, we get 2/4. These two fractions, 1/2 and 2/4, are equivalent because they represent the same amount of the whole. Let’s break it down: 1/2 represents one out of two equal parts. 2/4 represents two out of four equal parts. Even though the numbers are different, the actual amount of the whole represented by both fractions is the same. You can picture this as cutting a pie in half and then cutting each half into two more pieces. You’ll have four pieces, and two of them represent the same amount as the original half! To create equivalent fractions, simply find a number you can multiply both the numerator and denominator by. You can use any number, as long as you apply it to both parts of the fraction. Here are some examples: 1/3 x 2/2 = 2/6 3/4 x 3/3 = 9/12 5/8 x 4/4 = 20/32 So, the key is to find a number that will make the fraction look different without changing its overall value. It’s like changing the size of the slices in your pie, but making sure you still have the same amount of pie! See more new information: barkmanoil.com ### What Fraction Is Equivalent To -3/2? Okay, let’s dive into the world of fractions and figure out what fractions are equivalent to -3/2. First, let’s remember what “equivalent” means when it comes to fractions. It means they represent the same value, even if they look different. Think of it like this: cutting a pizza into 6 slices and taking 3 is the same as cutting a pizza into 12 slices and taking 6 – you get the same amount of pizza! Now, let’s get back to -3/2. This is a negative fraction, and it’s called an improper fraction because the numerator (the top number) is bigger than the denominator (the bottom number). But don’t worry, we can make this look simpler. The key is to understand that multiplying or dividing both the numerator and denominator of a fraction by the same number doesn’t change its value. It’s like resizing a picture – you change the dimensions, but the image remains the same. So, let’s find some equivalent fractions to -3/2: Multiply by 2: (-3 x 2) / (2 x 2) = -6/4. Multiply by 3: (-3 x 3) / (2 x 3) = -9/6. Multiply by 4: (-3 x 4) / (2 x 4) = -12/8. You can keep going, multiplying by any number! Now, let’s look at another way to find equivalent fractions. We can express -3/2 as a mixed number. This is a number that has a whole number part and a fraction part. To do this, we divide the numerator by the denominator: -3 ÷ 2 = -1 with a remainder of -1. This means -3/2 is the same as -1 1/2. Remember, finding equivalent fractions is like finding different ways to say the same thing. It’s important to understand the concept because it helps you solve problems and compare fractions more easily. Let’s talk about some applications of equivalent fractions: Adding and subtracting fractions: You can only add or subtract fractions if they have the same denominator. Finding equivalent fractions with a common denominator helps you do this. Comparing fractions: It’s easier to compare fractions if they have the same denominator. Simplifying fractions: You can simplify a fraction by dividing both the numerator and denominator by their greatest common factor. Let’s go over some FAQs about equivalent fractions. 1. Can I simplify -3/2? No, you can’t simplify -3/2 in the traditional way because the numerator and denominator don’t share any common factors other than 1. However, you can express it as a mixed number, which is a simpler way to represent the value. 2. How do I find the equivalent fraction for a given fraction? To find an equivalent fraction, multiply or divide both the numerator and denominator by the same number. Remember, you’re basically resizing the fraction without changing its value. 3. Why are equivalent fractions important? Equivalent fractions are crucial for understanding and working with fractions. They allow you to compare, add, subtract, and simplify fractions more easily. 4. Can I have a negative equivalent fraction? Yes, you can have negative equivalent fractions. Remember, multiplying or dividing both the numerator and denominator by a negative number will also give you an equivalent fraction. 5. Are all fractions equivalent to -3/2 negative? No, not all fractions equivalent to -3/2 are negative. For example, if you multiply both the numerator and denominator by -1, you get 3/(-2), which is equivalent to -3/2, but is not negative. Understanding equivalent fractions is essential in working with fractions and can make your life easier when dealing with these mathematical concepts. Remember, it’s all about finding different ways to represent the same value! ### Equivalent fractions (video) | Fractions | Khan Academy Yes. Equivalent fractions are interchangeable in every way, so they are a useful way of simplifying equations. The fraction 1/5 is equivalent to the fraction 12589/62945, but it’s much easier Khan Academy ### Equivalent Fractions – Math is Fun Equivalent Fractions. Equivalent Fractions have the same value, even though they may look different. These fractions are really the same: 1 2 = 2 4 = 4 8. Why are they the same? Math is Fun ### What are Equivalent Fractions? Definition, Methods & Examples The two equivalent fractions for $\frac{3}{8}$ are $\frac{6}{16}$ and $\frac{9}{24}$. Divide the numerator and denominator by the same number. We can divide the numerator and SplashLearn ### Equivalent fractions review (article) | Khan Academy No, a fraction is only an equivalent fraction when it can be obtained by multiplying or dividing both the numerator and denominator by the same number. This results in a set Khan Academy ### Equivalent fractions – Math.net How to find equivalent fractions. Any given fraction has an infinite number of equivalent fractions. We can find equivalent fractions by multiplying or dividing both the numerator Math.net ### Equivalent Fractions | ChiliMath There are two ways we can show why these fractions are equivalent using some arithmetic. One way is to start with ${2 \over 5}$ and multiply its top and bottom by $3$ to get the target fraction ChiliMath ### Equivalent Fractions Explained—Definitions, Examples, First, let’s start with the equivalent fractions definition: Math Definition: Equivalent Fractions. Equivalent fractions are fractions that have the same value but do Mashup Math ### Equivalent Fractions | Definition, Examples, Finding, Find Equivalent Fraction . Equivalent fractions also can be determined by multiplying or dividing by the numerator and the denominator by the same number of values. When we transform equivalent fractions into their Helping with Math ### Equivalent fractions and comparing fractions | 4th grade – Khan Academy In this lesson, you’ll learn all about equivalent fractions and how to compare them. With the help of models, number lines, and benchmark fractions, you’ll be a fraction master in no khanacademy.org Equivalent Fractions | Math With Mr. J Equivalent Fractions | Maths | Easyteaching

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Later that winter, during a routine cave survey, New York State biologists found thousands of dead bats in a limestone cave near Albany, many encrusted with a strange white fuzz. During the winters that followed, dead bats piled up in caves throughout the Northeast. The scientists would emerge filthy and saddened, with bat bones—each as thin and flexible as a pine needle—wedged into their boot treads. By the end of 2008, wildlife-disease researchers had identified the fuzz as a fungus new to North America. Today the fungus has spread to 19 states and 4 Canadian provinces, and infected nine bat species, including the endangered Indiana and gray bats. A 2010 study in the journal Science predicted that the little brown bat—once one of the most common bat species in North America—may go extinct in the eastern United States within 16 years. “When it first hit, I thought, ‘OK, is there anything we can do to keep it within this cave?’” remembers Hicks. “The next year it was, ‘Is there anything we can do to secure our largest colonies?’ And then the next year it was, ‘Can we keep any of these colonies going?’ Now we’re asking if we can keep these species going.” G. destructans also infects bats in Europe—but it doesn’t kill them, at least not in large numbers. G. destructans may have swept through European caves in the distant past, leaving only bats that could withstand the fungus. Researchers don’t know when and how the fungus made its way to North America, but they speculate that it may be so-called “pathogen pollution,” the inadvertent human transport of diseases—in this case possibly by a cave-visiting tourist—into new and hospitable habitats. With their undeserved association with creepy folk tales, bats don’t have much of a constituency. But bat biologists say the consequences of the North American die-off stretch far beyond the animals themselves. For instance, one million bats—the number already felled by white-nose syndrome—consume some 700 tons of insects, many of them pests, every year. Fewer bats mean more mosquitoes, aphids and crop failures. A study published in Science this spring estimated that bats provide more than $3.7 billion in pest-control services to U.S. agriculture every year. With G. destructans reaching farther each winter, Barton, Slack and an array of other biologists are racing to understand the fungus in time to contain it. Since scientists aren’t sure how easily people may spread the fungus, many caves have been closed, and tourists, recreational cavers as well as scientists are advised to clean their gear between trips underground. Barton and her students have shown that common cleaning products, such as Woolite and Formula 409, kill G. destructans without harming caving gear. But even as Barton, Slack and their colleagues patrol the perimeter of the disease, they acknowledge that the syndrome is likely to continue its spread across the continent. “Who’s going to live, and who’s going to die?” asks DeeAnn Reeder. “That’s the big thing I think about all the time.” Reeder, a biology professor at Bucknell University in central Pennsylvania, spends her days surrounded by white-nose syndrome. G. destructans thrives in nearby caves and mines, on many of the bats in her campus laboratories, and even on a set of petri dishes secured in an isolated laboratory refrigerator. Up close, the epidemic is more complicated than it first appears, for some bat species—and some individual bats—are proving more resistant than others. Reeder wants to know why. Reeder never expected to study white-nose syndrome, but like Barton, she was perfectly prepared for the job. Fascinated by mammals since her childhood summers in the Sierra Nevada, she studied primate physiology and behavior before switching to bats. At first, the reasons were practical—bats were easy to catch and sample in large numbers—but “I just fell in love with them,” Reeder says. “They’re so tough. I’ve always said that nothing will take them down, that they’re completely resilient. And then we got this fungus,” she says, shaking her head. “It caught us all off guard—and it caught them off guard, too.” After Reeder came to Pennsylvania in 2005, she outfitted her laboratory with a set of climate-controlled chambers designed to mimic natural cave conditions. She and her students had just begun to collect data on bat hibernation patterns when white-nose syndrome emerged. Suddenly, biologists all over the continent had questions about how bats behaved during hibernation, and Reeder was one of the only researchers well-positioned to answer them. “They’d say, ‘What do we know about hibernation?’ and I’d say, ‘Well, we know this much,’” says Reeder, holding a finger and thumb close together.

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Nativism, Nativist, Nativists Opposition to immigration on the grounds that an influx of foreigners will marginalize the English language, undermine American culture, destabilize American politics, and weaken the economic status of American workers. Anti-immigrant and anti-Catholic sentiment most prevalent during the decades leading up to the Civil War, when large numbers of Irish and German immigrants poured into the U.S. A xenophobic policy (or ideology) which stresses the interests of a country's native inhabitants over those of immigrants. Many (though not all) white Anglo-Saxon Protestants of nineteenth-century America became embodiments of this philosophy, to varying degrees. Those most concerned about foreign immigration joined nativist political parties. The most prominent of these parties was the American Party (a.k.a. the Native American Party, a.k.a. the Know Nothings), which began in 1843 and called for a 25-year residency qualification for citizenship and sought to elect only native-born Americans to political office. The Know Nothings enjoyed political victories on state and local levels, notably in Massachusetts and Delaware in 1854 (the pinnacle-year of their success). But the slavery issue eclipsed the nativists in importance and public attention, and ultimately divided their membership along sectional lines. Opposition to immigration based on fears that an influx of foreigners will marginalize the English language, undermine American culture, destabilize American politics, and weaken the economic status of American workers. The adoption or assimilation of American culture by foreign immigrants. The process through which a person attains citizenship in a country other than the one in which he or she was born. The process by which an immigrant becomes an American citizen. A racist and pejorative term for low-skilled, low-wage Asian laborers. "new Immigration", New Immigrants, New Immigrant Large-scale immigration to the United States from the nations of Eastern and Southern Europe between about 1880 and 1930. "old Immigration", Old Immigrant, Old Immigrants Immigration to the United States from Northwestern Europe—primarily from Great Britain, Ireland, and Germany—that occurred before 1880. e Pluribus Unum A Latin phrase meaning, "out of many, one," e pluribus unum has appeared on the Great Seal of the United States since 1782. Anarcho-syndicalism, Anarcho-syndicalist, Anarcho-syndicalism A radical philosophy of trade unionism that advocates seizure of control over industry by rank and file workers. Anarchist, Anarchists, Anarchism A believer in anarchism, the philosophy that holds that abolition of all forms of government is necessary to achieve true liberty. Robber Baron, Robber Barons A pejorative term used by some workers to describe the wealthy tycoons who built vast fortunes in the nineteenth century railroad, steel, and petroleum industries. The status of being a citizen of the United States. Citizenship can be attained either through birthright (for those born in the United States) or through naturalization (for those who immigrate to the United States from foreign countries). A person who flees from his or her home to another country out of fear of violence or persecution for political, religious, or ethnic reasons. The process by which an individual renounces citizenship in his or her country of birth. The Expatriation Act of 1907 forced American women who married foreign men to forfeit their American citizenship. Fear or hatred of foreigners. Communist philosophy rooted in the doctrines of Karl Marx as interpreted by V.I. Lenin. The government-sanctioned expulsion of an alien from the country. Assimilation, Assimilated, Assimilate The process through which a particular immigrant group abandons its ethnic traditions to adopt the cultural mores of mainstream America. Eugenics, Eugenicist, Eugenicists The idea that humanity can be improved through the selective breeding of those with superior genetic traits. Eugenics was a popular pseudo-scientific movement of the late nineteenth and early twentieth centuries, as people just becoming familiar with Charles Darwin's theories of evolution sought to intervene in normal human patterns of reproduction in order to advance supposedly desirable genetic traits and weed out undesirable ones. In practice, eugenics was often deeply racist, and it was sometimes used to justify atrocities such as the forced sterilization of people deemed genetically inferior. The theories of Aryan racial supremacy that underpinned German Nazism were rooted, in part, in eugenicist ideas, and the murderous evil of the Holocaust permanently discredited the eugenicist movement after World War II

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# Draw a triangle ABC with side BC = 7 cm, B = 45, A = 105. Then, construct a triangle whose sides are times the corresponding side of ΔABC. ANDConstruct a triangle of sides 4 cm, 5cm and 6cm and then a triangle similar to it whose sides areof the corresponding sides of the first triangle.does both questions have same method? pls clarify it!!! ∠B = 45°, ∠A = 105° Sum of all interior angles in a triangle is 180°. ∠A + ∠B + ∠C = 180° 105° + 45° + ∠C = 180° ∠C = 180° − 150° ∠C = 30° The required triangle can be drawn as follows. Step 1 Draw a ΔABC with side BC = 7 cm, ∠B = 45°, ∠C = 30°. Step 2 Draw a ray BX making an acute angle with BC on the opposite side of vertex A. Step 3 Locate 4 points (as 4 is greater in 4 and 3), B1, B2, B3, B4, on BX. Step 4 Join B3C. Draw a line through B4 parallel to B3C intersecting extended BC at C'. Step 5 Through C', draw a line parallel to AC intersecting extended line segment at C'. ΔA'BC' is the required triangle. Justification The construction can be justified by proving that In ΔABC and ΔA'BC', ∠ABC = ∠A'BC' (Common) ∠ACB = ∠A'C'B (Corresponding angles) ∴ ΔABC ∼ ΔA'BC' (AA similarity criterion) … (1) In ΔBB3C and ΔBB4C', ∠B3BC = ∠B4BC' (Common) ∠BB3C = ∠BB4C' (Corresponding angles) ∴ ΔBB3C ∼ ΔBB4C' (AA similarity criterion) On comparing equations (1) and (2), we obtain This justifies the construction. • 76 What are you looking for?

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# Algebraic expressions activity pdf PRACTICE: Translating Phrases to Algebraic Expressions Translate the sentences to algebraic equations. 1. The sum of a number and 16 is equal to 45. _____ 2. The product of 6 and m is 216. _____ 3. The difference of 100 and x is 57. ...Algebraic Expressions 2 Activity A. Convert the following word statements into algebraic expressions: 1. Six times a number increased by two times a different number. _____ 2. Three divided by a number added to eight times a different number. _____ 3. Nine times a number subtracted from another number. _____ B. Identify the parts of the ... Download. Full PDF Package. This paper. A short summary of this paper. 13 Full PDFs related to this paper. Read Paper. Lesson Plan Lounge Grade 7 Algebraic Expressions Standard 1.0- Knowledge of Algebra, Patterns, and Functions Topic B: Expressions, Equations, and Inequalities Indicator: 1. Write and evaluate expressions Objective: b.Writing and evaluating expressions worksheet pdf answer key. Algebraic Expressions Worksheets Math Fun Worksheets. Worksheets for writing expressions with Algebraic Expression Homework from verbal expressions Use the generator to customize the worksheets as a wish. ALGEBRA TRIVIA Reciprocal Fractions WorkSheetAlgebraic Expressions 2 Activity A. Convert the following word statements into algebraic expressions: 1. Six times a number increased by two times a different number. _____ 2. Three divided by a number added to eight times a different number. _____ 3. Nine times a number subtracted from another number. _____ B. Identify the parts of the ... Algebraic expressions grade 6 worksheets pdf.All measurements are in centimeters. Algebra worksheets printable. Three more than a number x 3 the quotient of a number and 8 y 8 six times a number 6 x n or 6n 15 less than a number z 15 the quotient of 30 and a number plus 10 30 x 10.1.1 Lesson 4 Chapter 1 Expressions and Number Properties Study Tip You can write the product of 4 and n in several ways. 4 ⋅ n 4n 4(n)Key Vocabulary numerical expression, p. 4 algebraic expression, p. 4 evaluate, p. 4 A numerical expression contains only numbers and operations. AnThese Order of Operations Worksheets will produce Algebraic problems for practicing Order of Operations calculations. You may change this if you wish, select the degree of difficulty to be either Easy (Four Numbers and Three Operations) or Hard (Five Numbers and Four Operations). You may introduce positive, negative, or mixed integers.©X sKJuBtnaX US3oDfdtJwIaXruen DLbLtC J.k l CAQlGlz nr0i4gIhhtZsS lr SeXsDeurzv4eTdz.L T BMCaLdPek 4wji rtDhe uI3nmfYiSnsiWt0ek zAolZg5eUbyr4aF A1F.a-3-Worksheet by Kuta Software LLC Answers to Evaluating Algebraic Expressions When the Variable is Known (ID: 1) distributive law wherever applicable. The pdf worksheets for grade 6 and grade 7 are split into two levels based on the difficulty involved. Pre-Algebra Worksheets | Algebraic Expressions Worksheets These Algebraic Expressions Worksheets will create algebraic statements for the student to simplify. You may select from 2, 3, or 4 terms Oct 06, 2021 · Download free printable worksheets algebraic expressions pdf of cbse and kendriya vidyalaya schools as per latest syllabus in pdf cbse class 7 maths worksheet algebraic expression 2. Question from very important topics are covered by ncert exemplar class 7 you also get idea about the type of questions and method to answer in your class 7th ... 7th Grade Simplifying Algebraic Expressions Worksheets Pdf. Fayette Martins. August 27, 2021. August 27, 2021. Employ this 7th grade free pdf worksheet to find the perimeter of quadrilaterals with dimensions expressed in algebraic expressions. When addition or subtraction signs separate an algebraic expression into parts.Free printable worksheets (pdf) with answer keys on Algebra I, Geometry, Trigonometry, Algebra II, and Calculus Download printable Algebraic Expressions Class 7 Worksheets in pdf format, Class 7 Maths Algebraic Expressions Worksheet has been prepared as per the latest syllabus and exam pattern issued by CBSE, NCERT and KVS. Also download free pdf Algebraic Expressions Class 7 Assignments and practice them daily to get better marks in tests and exams for Grade 7.worksheets in the category writing algebraic expressions some of the worksheets displayed are variable and verbal expressions writing basic algebraic expressions writing basic algebraic expressions work 1 write the expression or equation algebraically translating key words and phrases into algebraic expressions algebraic and numeric, writing ... ## The hydrocarbon c4h8 was burnt in air Assist your child with algebra and review the order of operations: division and multiplication before addition and subtraction, left to right. In this fall-themed worksheet version of the popular game two truths and one lie, learners will solve three equations and determine which two equations are true and which one is false. Pre algebra simplifying algebraic expressions worksheets pdf. The worksheets can be made either as pdf or html files the latter are editable in a word processor. 4a 2 5 2. All measurements are in centimeters. Work out the value of x. 2a 5b 3 a 2 5. B work out the value of r if the perimeter is 49 cm. ### Vela de parapente precios Algebraic expressions and integers downloadable worksheet. Working with algebraic expressions is a fundamental skill in algebra. This page provide printables on adding and subtracting algebraic expressions, multiplying and dividing algebraic expressions, simplifying algebraic expressions, learn the order of operations in algebra and the distributive property of multiplication.

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# 10 3 Slope and Rate of Change Learn • Slides: 12 10 -3 Slope and Rate of Change Learn to find rates of change and slopes. 10 -3 Slope and Rate of Change Vocabulary rate of change slope 10 -3 Slope and Rate of Change The rate of change of a function is a ratios that compares the difference between two output values to the difference between the corresponding input values. 10 -3 Slope and Rate of Change Additional Example 1 A: Using A Table to identify Rates of Change Tell whether the rates of change are constant or variable. +2 +1 +2 +3 Find the difference x 2 4 7 8 10 between y 5 11 20 23 29 consecutive data points. +6 +3 +6 +9 Find each ratio of the change in y to the change in x. The rate of change is constant. 10 -3 Slope and Rate of Change Caution! Be careful to put the difference in y-values in the numerator and the differences in x -values in the denominator when you write a rate of change. 10 -3 Slope and Rate of Change Additional Example 1 B: Using A Table to identify Rates of Change Tell whether the rates of change are constant or +1 +1 variable. Find the difference x 0 1 2 3 4 between y 0 3 5 8 10 consecutive data points. +2 +3 +3 +1 Find each ratio of the change in y to the change in x. The rates of change are variable. 10 -3 Slope and Rate of Change Check It Out: Example 1 Tell whether the rates of change are constant or variable. +3 +1 +2 +3 Find the difference x 0 2 5 6 9 between y 5 15 30 35 50 consecutive data points. +15 +10 +15 +5 Find each ratio of the change in y to the change in x. The rate of change is constant. 10 -3 Slope and Rate of Change When the rate of change is constant, the segments form a straight line. The constant rate of change of a line is its slope. 10 -3 Slope and Rate of Change Reading Math Recall that a function whose graph is a straight line is a linear function. 10 -3 Slope and Rate of Change Additional Example 2: Driving Application The table shows the driving distances that Jesse recorded. A. Determine whether the rates of change are constant or variable. The rate of change is constant. 10 -3 Slope and Rate of Change Additional Example 2: Driving Application B. Graph the data and connect the points with line segments. If the rate of change is constant, find and interpret the slope. The rate of change between 3 any two points is 5. The slope of the line is 3. 5 The slope is 3. This means he drove 3 mi. every 5 min. 5 10 -3 Slope and Rate of Change Check It Out: Example 2 The table shows the driving distances that Barry recorded. Time (min) 1 3 6 9 12 Distance (miles) 3 6 12 18 24 Determine whether the rates of change are constant or variable. 3 =3 1 6 =2 3 12 = 2 6 18 = 2 9 The rates of change are variable. 24 = 2 12

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Stratovolcanoes tend to form at subduction zones, or convergent plate margins, where an oceanic plate slides beneath a continental plate and contributes to the rise of magma to the surface. At rift zones, or divergent margins, shield volcanoes tend to form as two oceanic plates pull slowly apart and magma effuses upward through the gap. Volcanoes are not generally found at strike-slip zones, where two plates slide laterally past each other. “Hot spot” volcanoes may form where plumes of lava rise from deep within the mantle to the Earth’s crust far from any plate margins. Or click Continue to submit anonymously:

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# 9/12 Simplified, Simplify the Fraction 9/12 to Simplest Form The fraction 9/12 simplified is equal to 3/4. Here we learn to write 9/12 (9 divided by 12) in its simplest form, that is, we will reduce the fraction 9/12 to its lowest terms. Note that Let us now simplify the fraction 9/12 in its simplified form which is 3/4. There are two ways to simplify any fraction. 1. Divide both the numerator and the denominator of the given fraction by the greatest common divisor of the top and the bottom. The resultant fraction will be the simplified form of the given fraction. 2. Write the prime factorisations of the numerator and the denominator of the given fraction. Then cancel the common numbers appeared in the top and the bottom to reduce the fraction to its lowest terms. ## Simplify 9/12 to its Simplest Form In order to simplify the fraction 9/12, we will follow below steps. To simplify 9/12 in its simplest form, let us divide both the numerator 9 and the denominator 12 by their greatest common divisor (GCD). Now, we will find the GCD of 9 and 12. • The divisors of 9 are 1, 3, and 9. • The divisors of 12 are 1, 2, 3, 4, 6, and 12. Therefore, 3 is the highest number that divides both 9 and 12. So the greatest common divisor of 9 and 12 is 3. That is, GCD (9, 12) = 3. Divide the top and the bottom by GCD (9, 12) = 3. $\dfrac{9}{12} = \dfrac{9 \div 3}{12 \div 3} = \dfrac{3}{4}$. So the fraction 9/12 simplified in its simplest form is equal to 3/4. ## How to Simplify the Fraction 9/12 The video solution on how to simplify 9/12 is provided below: ## Reduce 9/12 to its Lowest Term To reduce 9/12 in its lowest terms, at first we write down the prime factorisations of 9 and 12. 9 = 3 × 3 12 = 2 × 2 × 3 Therefore, $\dfrac{9}{12} = \dfrac{3 \times 3}{2 \times 2 \times 3}$ ⇒ $\dfrac{9}{12} = \dfrac{3 \times \cancel{3}}{2 \times 2 \times \cancel{3}}$, here we cancel the common number 3. ⇒ $\dfrac{9}{12} = \dfrac{3}{2 \times 2} = \dfrac{3}{4}$ So the fraction 9/12 reduced in its lowest term is equal to 3/4. ## FAQs Q1: What is 9/12 simplified? Answer: 9/12 simplified as a fraction is equal to 3/4. Q2: How simplify 9/12 in its simplest form? Answer: The greatest common divisor (GCD) of 9 and 12 is GCD(9, 12) = 3. To simplify 9/12, divide the top 9 and the bottom 12 by their GCD. Thus, 9/12 = (9 ÷ 3)/(12 ÷ 3) = 3/4. So 3/4 is the simplest reduced form of the fraction 9/12. Share via:

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New species can arise in just a few thousand years, say scientists who have tracked the process in action. They say a DNA analysis has established that one type of Australian sea star has separated off in as little as six thousand years. "That's unbelievably fast compared to most organisms," says Rick Grosberg, professor of evolution and ecology at UC Davis. He and his colleagues studied two closely related 'cushion stars', Cryptasperina pentagona and C. hystera, living on the Australian coast. They're identical in appearance but live in different regions, with Hystera occuring on a few beaches and islands at the far southern end of the range of pentagona. And their sex lives are very, very different. Pentagona has male and female individuals that release sperm and eggs into the water where they fertilize, grow into larvae and float around for a few months before settling down and developing into adult sea stars. Hystera, by contrast, are hermaphrodites that brood their young internally and give birth to miniature sea stars ready to grow to adulthood. "It's as dramatic a difference in life history as in any group of organisms," says Grosberg. The researchers looked at the diversity in DNA sequences from sea stars of both species and estimated the length of time since the species diverged. And the results show that the species separated about 6,000 to 22,000 years ago - ruling out some ways in which new species evolve. For example, they clearly didn't diverge slowly with genetic changes over a long period of time, but were isolated quickly. Over the last 11,000 years, the boundary between cold and warm water in the Coral Sea has fluctuated north and south. A small population of the ancestral sea stars, perhaps even one individual, might have colonized a remote area at the southern end of the range and then been isolated by one of these changes in ocean currents, Grosberg says.

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Presentation is loading. Please wait. # Vector Operations Chapter 3 section 2 A + B = ? B A. ## Presentation on theme: "Vector Operations Chapter 3 section 2 A + B = ? B A."— Presentation transcript: Vector Operations Chapter 3 section 2 A + B = ? B A Vector Dimensions When diagramming the motion of an object, with vectors, the direction and magnitude is described in x- and y- coordinates simultaneously. This allows vectors to be used for 1-d and 2-d motion. How can I get to the red dot starting from the origin and can only travel in a straight line? x y There are 3 main different ways that I can travel to get from the origin to the red dot by only traveling in a straight lines. x y Solving For The Resultant of 2 Perpendicular Vectors When two vectors are perpendicular to each other it forms a right triangle, when the resultant is formed. Right triangles have special properties that can be used to solve specific parts of the triangle. Such as the length of sides and angles. (length of hypotenuse)²=(length of leg)²+(length of other leg)² Magnitude of a Vector To determine the magnitude of two vectors, the Pythagorean Theorem can be used As long as the vectors are perpendicular to each other. Pythagorean Theorem c²=a²+b² (length of hypotenuse)²=(length of leg)²+(length of other leg)² Applied Pythagorean Theorem c2=a2+b R²=Δy²+Δx² (Mathematics) (Physics) Δx Δy R a b c Direction of a Vector To determine the direction of the vector, use the tangent function. Tangent Function Tanθ=opp/adj opp θ adj Applied Tangent Function Δx Δy R a=opp b=adj c θ θ 𝑇𝑎𝑛 𝜃= Δ𝑦 Δ𝑥 (Physics) 𝑇𝑎𝑛 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗 (Mathematics) Recall Vector Properties Δx Δy R Δx Δy R = θ θ Example Problem A soldier travels due east for 350 meters then turns due north and travels for another 100 meters. What is the soldiers total displacement? Example Picture Example Work Example Answer R= ° Vector Components Every vector can be broken down into its x and y components regardless of its magnitude or direction. Vectors Pointing Along a Single Axis When a vector points along a single axis, the second component of motion is equal to zero. Vectors That Are Not Vertical or Horizontal Ask yourself these questions. How much of the vector projects onto the x-axis? How much of the vector projects onto the y-axis? Components of a Vector x y A A x θ A x Resolving Vectors into Components Components of a vector – The projection of a vector along the axis of a coordinate system. x-component is parallel to the x-axis y-component is parallel to the y-axis These components can either be positive or negative magnitudes. Any vector can be completely described by a set of perpendicular components. Vector Component Equations Solving for the x-component of a vector. 𝐴𝑥=𝐴𝑐𝑜𝑠𝜃 (𝑥−𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 • cos⁡(𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴) Solving for the y-component of a vector. 𝐴𝑦=𝐴𝑠𝑖𝑛𝜃 Example Problem Break the following vector into its x- and y- components. A = ° Example Problem Work A = ° Example Problem Answer Ax = 4.66 m/s Ay = 3.78 m/s Example Problem: A plane takes off from the ground at an angle of 15 degrees from the horizontal with a velocity of 150mi/hr. What is the horizontal and vertical velocity of the plane? Example Picture Example Work Example Answer Horizontal velocity = 144.89 miles per hour Vx=144.89mi/hr Vertical velocity = miles per hour Vy=38.82mi/hr Adding Non-Perpendicular Vectors When vectors are not perpendicular, the tangent function and Pythagorean Theorem can’t be used to find the resultant. Pythagorean Theorem and Tangent only work for two vectors that are at 90 degrees (right angles) Non-Perpendicular Vectors To determine the magnitude and direction of the resultant of two or more non-perpendicular vectors: Break each of the vectors into it’s x- and y- components. It is best to setup a table to nicely organize your components for each vector. Component Table x-component y-component Vector A - (A) Vector B - (B) Vector C - (C) Add more rows if needed Resultant - (R) Non-Perpendicular Vectors Once each vector is broken into its x- and y- components : The components along each axis can be added together to find the resultant vector’s components. Rx = Ax + Bx + Cx + … Ry = Ay + By + Cy + … Only then can the Pythagorean Theorem and Tangent function can be used to find the Resultant’s magnitude and direction. Example Problem During a rodeo, a clown runs 8.0m north, turns 35 degrees east of north, and runs 3.5m. Then after waiting for the bull to come near, the clown turns due east and runs 5.0m to exit the arena. What is the clown’s total displacement? Practice Problem Picture Step #1: Draw a picture of the problem Practice problem Work Step #2: Break each vector into its x- and y- components. x-component y-component Vector A - (A) Vector B - (B) Vector C - (C) Resultant - (R) Step #3: Find the resultant’s components by adding the components along the x- and y-axis. x-component y-component Vector A - (A) Vector B - (B) Vector C - (C) Resultant - (R) + Step #4: Find the magnitude of the vector by using the Pythagorean theorem. R2 = Δx2 + Δy2 Step #5: Find the direction of the vector by using the tangent function. Tan θ = Δy/Δx Practice Problem Answer Step #5: Complete the final answer for the resultant with its magnitude and direction. Practice Problem Answer Resultant displacement = 57.21º Download ppt "Vector Operations Chapter 3 section 2 A + B = ? B A." 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When the operational form of the journalistic process is executed in a different format, it will be interpreted differently. This is contributable to the effect that the online interface is creating a new form of journalism, which uses different protocols to communicate it’s message. Three key parts, as discussed by Mark Deuze (2003), are: 1. the ability to become hypertextual (i.e. create content within content by the use of hyperlinks to other sources); 2. interactivity (i.e. adding social features to the content/the webpage’s interface) and 3. multimedia (i.e. adding film, photo and sound in order to provide extra information on the given subject, or add aesthetic value to the text). Because of these singular abilities online, ‘non-traditional’ features have become prominent to the role of designing the form and content of an online news source. In the case of Deuze these are: Adaptive interactivity, navigational and functional. As explained in chapter one, a medium, especially nowadays, cannot operate in a singular form. Interactive media communicate through different platforms, giving the user the possibilities to switch and change his or her focus amongst several differently designed interfaces. The notion that a medium nowadays is not able to function in isolation changes the design of the interface and instigates new procedures in order to support the users need for social interaction. For a medium to gain legitimacy there needs to be a point of recognition, in which the user recognizes and accepts references he/she has experienced earlier, seen by this user as other trustworthy media. The elements used to trigger the memory of the user act as manipulator, which is meant to create an arbitrary symbolic manipulation, (re-)creating an appearance of authority. This forms a continuous reciprocity of hypermediacy, which in effect creates this authentic feeling. Creating something that will be transparent, which enforces it’s status of a trustworthy source of information, depends on a ‘cultural literacy’. This literacy isn’t always directly obvious when looking at the interface, as it occurs in specifically designed details (Phillips, 2010). In order to experience this transparency the user has to acknowledge and understand the literacy of the medium. In other words, media have to evoke hypermediacy through its interface, in order to prevent the users previous experience from becoming a liability. To implement what we have demonstrated in the previous chapter and apply this concept to a news source, we need to look at how users interact with the media and what the properties of this media’s authoritative role is.

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You are Here: Home >< Maths # Solving inequality watch Announcements 1. (Original post by around) Or you can do what OP originally suggested (as none of those expressions are ever negative) The LHS is and when surely the expression is negative. Also if the RHS is negative. 2. you will get (7x+1)(x+1)(x-1)>0 finally,then,look (1).suppose 7x+1>0,get x>-1/7,then(x+1)(x-1)>0;get x>1 & x<-1 the intersection of x>-1/7 and x>1 & x<-1 will be x>1 (2).suppose 7x+1<0,get x<-1/7,then(x+1)(x-1)<0;get -1<x<1 the intersection of x<-1/7 and -1<x<1 will be -1<x<-1/7 Union (1)and(2) will be the answer -1<x<-1/7 & x>1 this inequality will be more diffcult when replace > with >= 3. (Original post by steve2005) The LHS is and when surely the expression is negative. Also if the RHS is negative. Multiply by {x-1}^2 and {x+1}^2 ? I meant these. 4. (Original post by D-Day) Changing the sign only happens when dividing by a negative constant. Here you are multiplying by an expression. Not true. When you multiply through by variables you have to consider the cases when what you're multiplying by is negative. So, in this case, you'd have to consider cases where: 1. and , that is 2. and , that is 3. and , that is It makes no difference whether it's constant or not. TSR Support Team We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out. This forum is supported by: Updated: August 8, 2009 Today on TSR ### Should I drop out of uni ...to become a pro gamer? ### University open days • University of Buckingham Fri, 14 Dec '18 • University of Lincoln Mini Open Day at the Brayford Campus Undergraduate Wed, 19 Dec '18 • University of East Anglia Fri, 4 Jan '19 Poll Useful resources ### Maths Forum posting guidelines Not sure where to post? Read the updated guidelines here ### How to use LaTex Writing equations the easy way ### Study habits of A* students Top tips from students who have already aced their exams

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April 16, 2010 The very first exercise at Programming Praxis evaluated expressions given in reverse-polish notation. In today’s exercise, we write a function to evaluate expressions given as strings in the normal infix notation; for instance, given the string “12 * (34 + 56)”, the function returns the number 1080. We do this by writing a parser for the following grammar: |expr + term| |expr – term| |term * fact| |term / fact| |( expr )| The grammar specifies the syntax of the language; it is up to the program to provide the semantics. Note that the grammar handles the precedences and associativities of the operators. For example, since expressions are made up from terms, multiplication and division are processed before addition and subtraction. The normal algorithm used to parse expressions is called recursive-descent parsing because a series of mutually-recursive functions, one per grammar element, call each other, starting from the top-level grammar element (in our case, expr) and descending until the beginning of the string being parsed can be identified, whereupon the recursion goes back up the chain of mutually-recursive functions until the next part of the string can be parsed, and so on, up and down the chain until the end of the string is reached, all the time accumulating the value of the expression. Your task is to write a function that evaluates expressions according to the grammar given above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below. Pages: 1 2

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Updated by Sharon Suh on May 19, 2015 REPORT Sharon Suh Owner 7 items   2 followers   0 votes   9 views # Suh's Portfolio 14-15 Professional Portfolio for 2014-2015 My on-level math class will score at least 80% on post assessments for each math unit. Beginning of the year data: 12 out of 20 students failed 4th grade Math STAAR. Action Plan: 1) small group instruction/ intervention during class with support from inclusion teachers and instructional coach 2) after-school tutoring 1x/ week 1 ## Unit 1 Post Assessment Unit 1 - Expressions, Equations, & Volume • Solve multi-step story problems involving multiplication and division with remainders • Multiply and divide with multi-digit numbers • Demonstrate an understanding of volume using multiplication • Find all factor pairs for whole numbers between 1 and 100 15 out of 20 students scored at least 80% 5 students scored between 69 - 77% 2 ## Unit 2 Post Assessment Unit 2 Post Assessment - Adding & Subtracting Fractions • Add and subtract fractions with unlike denominators • Solve story problems involving addition and subtraction of fractions with unlike denominators • Find common denominators for fractions with unlike denominators • Find the greatest common factor and least common multiple to help simplify fractions and find common denoinators • Multiply multi-digit numbers 19 out of 20 students scored at least 80% . 1 student scored 72%. 3 ## Unit 3 Post Assessment Unit 3 Post Assessment - Place Value and Decimals - Divide a 3-digit whole number by a 2-digit whole number - Read, write, expand, order, model, and compare decimal numbers to the thousandths place - Round decimals to hundredths place - Multiply and divide whole and decimal numbers by 10 - Add and subtract decimal numbers to the hundredths place - Identify equivalent fractions and decimals 100% of students scored at least 80%. 4 ## Unit 4 Post Assessment Unit 4 Post Assessment - Multiply and Divide Whole Numbers and Decimals • Use a variety of strategies for multiplying and dividing multi-digit whole numbers • Practice using the standard algorithm to multiply multi-digit whole numbers • Begin multiplying and dividing with decimal numbers 19 out of 21 students scored at least 80% or higher. 2 students scored between 74-78%. 5 ## Unit 5 Post Assessment Unit 5 Post Assessment - Multiply and Divide Fractions • Multiply fractions by whole numbers (1/3 × 12 = 4) • Use rectangular arrays to show multiplication of a fraction by a fraction ( 1/3 × 3/4 =1/4) • Divide a whole number by a fraction (4 ÷ 1/3 = 12) • Divide a unit fraction (a fraction with a 1 in the numerator) by a whole number (1/3 ÷ 4 = 1/12) 17 out of 21 students scored at least 80% or higher. 4 students scored between 55 - 75%. 6 ## Unit 6 Post Assessment Unit 6 Post Assessment - Graphing, Geometry & Volume • Calculate the volume of a rectangular prism using a formula and other strategies • Graph points in the coordinate plane • Sort and classify triangles, quadrilaterals, and other 2-D shapes 20 out of 21 students scored at least 80% or higher. 1 student scored 60%. 7 ## Personal Finance Unit Personal Finance Unit • Define income tax, payroll tax, sales tax, and property tax • Tell the difference between gross income and net income • Explain the advantages and disadvantages of making payments with check, debit card, credit card, and electronic payments • Develop a system for tracking finances and using financial records • Describe actions that might be taken to balance a budget when expenses exceed income • Balance a simple budget 15 out of 21 students scored at least 80% or higher. 6 students scored between 66 - 77%.

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Impact Melt at Memory Bay April 05, 2008 This photo shows a meteor impact melt cliff found in the central peak area of the Manicouagan Impact Structure, Quebec, Canada. Approximately 214 million years ago, an estimated 10 k (6.2 mi) wide bolide impacted here at a velocity of between 12 and 30 k (7.4 and 18.6 mi) per second. The resultant 100 k (62 mi) diameter crater is one of the largest impact craters still preserved on the surface of the Earth. The water filled circular annular moat that's prominent on images taken from Earth orbit is only one third of the size of the original crater. This moat fills a ring where impact-brecciated rock was eroded away by glaciation. The illustrated impact melt cliff and talus (debris at the base of the cliff) is composed of target rock that was made temporarily molten from the energy released during the impact of the bolide. The heat released was so intense that's believed it took 1,600 to 5,000 years before the melted rocks cooled. There's no detectable meteorite component in the Manicouagan structure melt rock (Palme et al., 1978). This impact melt and talus outcrop is found in an inlet, cut into the central peak of the impact structure, known as Memory Bay. Since the impact, millions of years of erosion have created the existing landforms at the Manicouagan impact structure.

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# MATHEMATICS MCQs PDF- RRB JE EXAM 2024 ## MATHEMATICS MCQs PDF- RRB JE EXAM 2024 Hello Aspirants, 1. Basic Arithmetic: • Addition: The sum of two or more numbers. • Subtraction: Finding the difference between two numbers. • Multiplication: The product of two or more numbers. • Division: Dividing one number by another. 2. Algebra: • Variables and Constants: Variables represent unknown quantities, while constants have fixed values. • Equations: Mathematical statements that express the equality of two expressions. • Linear Equations: Equations of the form “ax + b = c,” where a, b, and c are constants. • Quadratic Equations: Equations of the form “ax^2 + bx + c = 0,” where a, b, and c are constants and x represents the variable. 3. Geometry: • Shapes and Figures: Triangles, circles, squares, rectangles, etc. • Area: The measure of the region enclosed by a shape. • Perimeter: The sum of all the sides of a shape. • Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 4. Trigonometry: • Trigonometric Functions: Sine (sin), Cosine (cos), Tangent (tan), etc. • Trigonometric Identities: Relationships between trigonometric functions. • Right Triangle Trigonometry: Using trigonometric functions to solve problems involving right-angled triangles. 5. Calculus: • Differentiation: Finding the derivative of a function. • Integration: Finding the integral of a function. • Derivative Rules: Power rule, product rule, chain rule, etc. 6. Probability and Statistics: • Probability: The likelihood of an event occurring. • Mean, Median, and Mode: Measures of central tendency. • Variance and Standard Deviation: Measures of dispersion. These are just a few key topics in mathematics. Depending on your specific needs or areas of interest, there are many more concepts to explore. ### Most Important Maths Question Answer Q.1. 1.75, 5.6, 7 का महत्तम समापवर्तक = ? (A) 0.7 (B) 0.07 (C) 3.5 (D) 0.35 Ans . D Q.2. 3, 2.7, 0.09 का लघुतम समापवर्त्य = ? (A) 2.7 (B) 0.27 (C) 0.027 (D) 27 Ans . D Q.3. रोहित ने रु 22.25 के 25 पैसे तथा 50 पैसे के कुल 52 टिकट ख़रीदे. इसमें उसने 50 पैसे वाले टिकटो पर कितनी राशी व्यय की ? (A) रु 12.50 (B) रु 14 (C) रु 18 (D) इनमे से कोई नहीं Ans . D Q.4. किसी संख्या का वर्ग दो संख्याओ 75.15 तथा 60.12 के वर्गों के अंतर के बराबर है. वह संख्या कोनसी है? (A) 46.09 (B) 48.09 (C) 45.09 (D) 47.09 Ans . C Q.5. दो संख्याओ का गुणनफल 0.008 है. इनमे से एक संख्या दूसरी संख्या के 1/5 के बराबर है. इनमे से छोटी संख्या क्या होगी? (A) 0.2 (B) 0.4 (C) 0.02 (D) 0.04 Ans . D Q.6. यदि A तथा B मिलकर किसी कार्य को 15 दिन में समाप्त करे तथा B अकेला इस कार्य को 20 दिन में समाप्त करे, तो A अकेला इस कार्य को कितने दिन में समाप्त करेगा ? (A) 60 दिन (B) 45 दिन (C) 40 दिन (D) 30 दिन Ans . A Q.7. A अकेला किसी कार्य को 18 दिन में तथा B अकेला इसे 15 दिन में पूरा कर सकता है. B अकेले ने इस पर 10 दिन में कार्य करके छोड़ दिया. शेष कार्य को A अकेला कितने और दिनों में पूरा करेगा? (A) 5 दिन (B) 7/2 दिन (C) 6 दिन (D) 8 दिन Ans . C Q.8. A और B मिलकर किसी कार्य को 8 दिन में पूरा कर सकते है. B अकेला उसी कार्य को 12 दिन में पूरा कर सकता है. B अकेला इस कार्य में पुरे 4 दिन लगा रहता है. इसके पश्चात् A अकेला उसे पूरा करने में कितने दिन और लेगा ? (A) 3 दिन (B) 16 दिन (C) 18 दिन (D) 20 दिन Ans . B Q.9. A तथा B मिलकर एक कार्य को 18 दिन में समाप्त कर सकते है, जबकि B तथा C मिलकर इसे 24 दिन में और C तथा A मिलकर इसे 36 दिन में समाप्त कर सकते है. A, B, C तीनो मिलकर इस कार्य को कितने दिन में समाप्त कर सकेंगे? (A) 15 दिन (B) 16 दिन (C) 17 दिन (D) 18 दिन Ans . B Q.10. A किसी कार्य को 18 दिन में, B, 20 दिन में तथा C, 30 दिन में पूरा कर सकता है. B तथा C मिलकर इस कार्य को आरंभ करते है किन्तु 2 दिन बाद वे कार्य छोड़कर चले जाते है. शेष कार्य को पूरा करने में A कितना समय लेगा? (A) 10 दिन (B) 12 दिन (C) 15 दिन (D) 16 दिन Ans . C ### Most Important Maths Question Answer Q 1 . कुछ व्यक्ति किसी काम को 15 दिन में करते है। यदि 15 व्यक्ति अधिक होते तो काम करने में 3 दिन कम लगते। तो शुरू में कितने व्यक्ति थे। (A). 45 (B). 50 (C). 55 (D). 60 Ans. 60 Q 2 . यदि एक दुकानदार एक प्रकार क चावल जिसका मूल्य 8 रू. प्रति किलो. तथा दूसरा प्रकार के चावल जिसका मूल्य 11 रूपये/किलो. के साथ मिलाकर मिश्रित मूल्य 9 रूपये/किलो. के भाव से बेचने पर उसको न लाभ न हानि है। उनका अपुपात ज्ञात करो? (A). 3:2 (B). 2:1 (C). 1:2 (D). 2:3 Ans. 2:1 Q 3 . 28 व्यक्ति किसी काम को 17 दिन में पूरा कर सकते हे यदि 6 व्यक्ति अधिक हो जाए तो वही काम कितने समय में समाप्त होगा ? (A). 12 दिन (B). 14 दिन (C). 15 दिन (D). 16 दिन Ans. 14 दिन Q 4 . कुछ व्यक्ति किसी काम को 45 दिन में करते है। यदि 5 व्यक्ति अधिक होते तो कमा करने में 5 दिन कम लगते। तो शुरू में कितने व्यक्ति थे ? (A). 40 (B). 45 (C). 50 (D). 35 Ans. 40 Q 5 . 8 बच्चों का एक समूह का औसत भार 25 किलो है। एक अध्यापक के आ जाने से उसके औसत भार 2 किलो बढ़ जाता है तो अध्यापक का वजन बताओ ? (A). 41 (B). 43 (C). 45 (D). 39 Ans. 43 Q 6 . A एक तिहाई काम 5 दिन में करता है। B शेष काम को 8 दिनों में पूरा करता है। तो A तथा B मिलकर काम को कितने समय में पूरा करेंगे ? (A). 9 3/5 दिन (B). 6 7/9 दिन (C). 6 2/3 दिन (D). 8 दिन Ans. 6 2/3 दिन Q 7 . A एक तिहाई काम 5 दिन में करता है। B शेष काम को 8 दिनों में पूरा करता है। तो A तथा B मिलकर काम को कितने समय में पूरा करेंगे ? (A). 9 3/5 दिन (B). 6 7/9 दिन (C). 6 2/3 दिन Ans. (D). 8 दिन Q 8 . किसी काम का 1/5 भाग 5 दिनों में कर सकता है, किसी काम का 2/5 भाग 8 दिनों में कर सकता है, तो और मिलकर कितनों दिनों काम समाप्त करेंगे ? (A). 9 1/14 दिनों (B). 11 1/9 दिनों (C). 13 दिनों (D). 12 दिनों Ans. 11 1/9 दिनों Q 9 . 31 पुरूष और 1 बच्चा मिलकर किसी काम को 6 दिनों में समाप्त कर सकते हैं। एक पुरूष काम को अकेले 10 दिनों में समाप्त कर सकता है, तो एक बच्चा अकेले काम को कितने दिनों में समाप्त करेगा ? (A). 16 दिन (B). 6 दिन (C). 4 दिन (D). 15 दिन Ans. 15 दिन Q 10 . A एक काम को 6 दिनों में तथा B 9 दिनों में कर सकता है। यदि वे 2 दिन तक एक साथ काम करते है, तो अब कितना हिस्सा काम बचा है ? (A). 3/9 (B). 4/9 (C). 13/18 (D). 7/18 Ans. 4/9 ### Most Important Maths Question Answer Q 1 . एक रेलगाड़ी किसी स्टेशन पर खड़े एक व्यक्ति की 20 सेकेण्ड में पार कर जाती। यदि रेलगाड़ी की चाल 20मी./से. हो तो रेलगाड़ी की लम्बाई ज्ञात करो? (A). 200मी. (B). 400मी. (C). 600मी. (D). 120मी. Ans. 400मी. Q 2 . 7 व्यक्तियों का औसत आयु 22 वर्ष है यदि 1 व्यक्ति उनके साथ मिल जाता है तो उनकी औसत आयु 1 वर्ष बढ़ जाता है तो जो व्यक्ति आया उसकी आयु कितना है? (A). 26 वर्ष (B). 28 वर्ष (C). 30 वर्ष (D). 32 वर्ष Ans. 30 वर्ष Q 3 . 5 मित्रों की औसत आयु 40 वर्ष है उनमें से एक मित्र चला जाता है तो अब औसत आयु 38 वर्ष हो जाता है। हटा गया मित्र की आयु कितना है ? (A). 44 वर्ष (B). 48 वर्ष (C). 40 वर्ष (D). 36 वर्ष Ans. 48 वर्ष Q 4 . यदि कोई ट्रेन 5 मिनट में 11 किमी. चलता है तो उस ट्रेन की चाल किमी./घंटा में कितना होगा? (A). 121 किमी./घंटा (B). 132 किमी./घंटा (C). 123 किमी./घंटा (D). 125 किमी./घंटा Ans. 132 किमी./घंटा Q 5 . एक ट्रेन एक प्लेटफार्म जिसकी लम्बाई 120 मीटर है उसे 10 सेकेण्ड में पार कर जाती है और प्लेटफार्म पर खड़े एक व्यक्ति को 65 सेकेण्ड में पार कर जाती है तो ट्रेन की चाल किमी./घंटा में बताओ? (A). 90 किमी./घंटा (B). 105 किमी./घंटा (C). 108 किमी./घंटा (D). 111 किमी./घंटा Ans. 108 किमी./घंटा Q 6 . स्थान A से B की दूरी को एक कार 50 किमी./घंटा की गति से जाता है और 30 किमी./घंटा की गति से आता है तो कार का औसत चाल ज्ञात कीजिए ? (A). 35 किमी./घंटा (B). 37.5 किमी./घंटा (C). 42.5 किमी./घंटा (D). 45 किमी./घंटा Ans. 37.5 किमी./घंटा Q 7 . क्रमागत 7 सम संख्याओं का औसत मान 24 है तो सबसे छोटी संख्या ज्ञात कीजिए ? (A). 18 (B). 22 (C). 16 (D). 20 Ans. 18 Q 8 . क्रमागत 7 सम संख्याओं का औसत मान 24 है तो सबसे छोटी संख्या ज्ञात कीजिए ? (A). 18 (B). 22 (C). 16 (D). 20 Ans. 18 Q 9 . एक समूह में 10 लड़को का औसत भार 21 किलो. है अगर प्रथम 5 लड़को का औसत भार 20 किलो हो और अगले 4 लड़को का औस भार 22 किलो हो तो अन्तिम लड़के का वजन कीजिए। (A). 20 किलो. (B). 21 किलो. (C). 22 किलो. (D). 23 किलो. Ans. 22 किलो. Q 10 . 17 छात्रों का औसत भार 17 किलो. है गलती से 19 और 21 के स्थान पर 11 तथा 12 जोड़ दिया जाता है तो उनका सही औसत भार क्या होगा ? (A). 15 किलो. (B). 16.5 किलो. (C). 17 किलो. (D). 17.5 किलो. Ans. 17 किलो.

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Both Positive and Negative Reinforcements Can Create Behavior Changes Positive and negative reinforcements are not as simple as they seem, and are often misunderstood. Usually mistaken for a system of reward or punishment, these terms refer to psychological processes that cause certain behaviors to be repeated. In the field of substance abuse, positive and negative reinforcements can be helpful at encouraging desirable behaviors or substituting new behaviors for unhealthy habits. When considering treatments, therapists should carefully understand how positive and negative reinforcements function – especially since research suggests people who abuse drugs or alcohol to avoid negative feelings are more likely to have lifelong problems with substance abuse than people who use drugs for the pleasurable feelings they provide. Reinforcement is any action that causes a behavior to be repeated more frequently. For example, a mouse will travel a maze over and over if he smells cheese at the end of the tunnel. Also called positive reinforcement, the subject receives a reward for good behavior and is likely to repeat it. Negative reinforcement also causes a behavior to be repeated, but in this case, the action causes a bad feeling or situation to go away. For example, some people repeatedly self-medicate with prescription drugs, alcohol or other substances because it removes unpleasant feelings of stress or anxiety. For people with substance abuse problems, the perceived rewards of drug abuse were probably learned long ago: taking a drug or consuming alcohol brings a feeling of pleasure or euphoria, however brief. Experts warn that when the good feelings wear off, the user is likely to keep abusing the drug because it brings relief from bad feelings – such as stress, anxiety or withdrawal. Unfortunately, this can create a difficult pattern of negative reinforcement and cause a patient to abuse substances for longer periods of time. Sometimes negative reinforcements cause a person to remain stuck even when the consequences aren’t likely to take place, which can open the door to compulsive behaviors. One example is if a person avoids social situations even as an adult because he received repeated embarrassment or ridicule as a child. The mind feels compelled to rely on old mechanisms or behavior patterns, even though they aren’t necessary anymore. Some patients with behavior disorders say they self-medicate with illegal substances because they provide quicker relief or escape from depression or pain than prescribed drugs. Referred to as “bait and switch,” the good feelings associated with drugs are the initial bait, but then the patient continues the substance abuse because it becomes the way bad feelings or bad situations are removed. There are treatment options for applying negative and positive reinforcements in substance abuse cases. Therapists can try eliminating the stressful situation that causes the patient to need to escape. This may mean counseling family members on how to be a positive influence on their loved ones, instead of berating them and causing more stress. Another strategy is to allow the patient to encounter the stressor, or literally face their fears, and then not permit them to resort to their escape strategy – but instead find new ways to cope. Teaching tolerance skills for dealing with a range of emotions may also be helpful, especially when people self-medicate. Positive and negative reinforcements can both bring success in substance abuse cases, especially when the substance problem is accompanied by anxiety or depression.

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Categories ## Mixture | Algebra | AMC 8, 2002 | Problem 24 Try this beautiful problem from Algebra based on Mixture from AMC-8, 2002. ## Mixture | AMC-8, 2002 | Problem 24 Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice? • 34% • 40% • 26% ### Key Concepts Algebra Mixture Percentage AMC-8, 2002 problem 24 Challenges and Thrills in Pre College Mathematics ## Try with Hints Find the amount of juice that a pear and a orange can gives… Can you now finish the problem ………. Find total mixture can you finish the problem…….. 3 pear gives 8 ounces of juice . A pear gives $\frac {8}{3}$ ounces of juice per pear 2 orange gives 8 ounces of juice per orange An orange gives $\frac {8}{2}$=4 ounces of juice per orange. Therefore the total mixer =${\frac{8}{3}+4}$ If She makes a pear-orange juice blend from an equal number of pears and oranges then percent of the blend is pear juice= $\frac{\frac{8}{3}}{\frac{8}{3}+4} \times 100 =40$ Categories ## Probability Problem | AMC 8, 2016 | Problem no. 21 Try this beautiful problem from Probability. ## Problem based on Probability | AMC-8, 2016 | Problem 21 A box contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? • $\frac{3}{5}$ • $\frac{2}{5}$ • $\frac{1}{4}$ ### Key Concepts probability combination fraction Answer: $\frac{2}{5}$ AMC-8, 2016 problem 21 Challenges and Thrills in Pre College Mathematics ## Try with Hints There are 5 Chips, 3 red and 2 green Can you now finish the problem ………. We draw the chips boxes in such a way that we do not stop when the last chip of color is drawn. one at a time without replacement Can you finish the problem…….. There are 5 Chips, 3 red and 2 green we draw the chips boxes in such a way that we do not stop when the last chip of color is drawn. if we draw all the green chip boxes then the last box be red or if we draw all red boxes then the last box be green but we draw randomly. there are 3 red boxes and 2 green boxes Therefore the probability that the 3 reds are drawn=$\frac{2}{5}$ Categories ## Pattern Problem| AMC 8, 2002| Problem 23 Try this beautiful problem from Algebra based on Pattern. ## Pattern – AMC-8, 2002- Problem 23 A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles? • $\frac{5}{9}$ • $\frac{4}{9}$ • $\frac{4}{7}$ ### Key Concepts Algebra Pattern Fraction Answer:$\frac{4}{9}$ AMC-8 (2002) Problem 23 Pre College Mathematics ## Try with Hints The same pattern is repeated for every $6 \times 6$ tile Can you now finish the problem ………. Looking closer, there is also symmetry of the top $3 \times 3$ square can you finish the problem…….. If we look very carefully we must notice that, The same pattern is repeated for every $6 \times 6$ tile Looking closer, there is also symmetry of the top $3 \times 3$ square, Therefore the fraction of the entire floor in dark tiles is the same as the fraction in the square Counting the tiles, there are dark tiles, and total tiles, giving a fraction of $\frac{4}{9}$. Categories ## Area of the Trapezoid | AMC 8, 2002 | Problem 20 Try this beautiful problem from Geometry based on Area of Trapezoid. ## Area of the Trapezoid – AMC- 8, 2002 – Problem 20 The area of triangle XYZ is 8  square inches. Points  A and B  are midpoints of congruent segments  XY and XZ . Altitude XC bisects YZ.What is the area (in square inches) of the shaded region? • $6$ • $4$ • $3$ ### Key Concepts Geometry Triangle Trapezoid Answer:$3$ AMC-8 (2002) Problem 20 Pre College Mathematics ## Try with Hints Given that Points  A and B  are midpoints of congruent segments  XY and XZ and Altitude XC bisects YZ Let us assume that the length of YZ=$x$ and length of $XC$= $y$ Can you now finish the problem ………. Therefore area of the trapezoid= $\frac{1}{2} \times (YC+AO) \times OC$ can you finish the problem…….. Let us assume that the length of YZ=$x$ and length of $XC$= $y$ Given that area of $\triangle xyz$=8 Therefore $\frac{1}{2} \times YZ \times XC$=8 $\Rightarrow \frac{1}{2} \times x \times y$ =8 $\Rightarrow xy=16$ Given that Points  A and B  are midpoints of congruent segments  XY and XZ and Altitude XC bisects YZ Then by the mid point theorm we can say that $AO=\frac{1}{2} YC =\frac{1}{4} YZ =\frac{x}{4}$ and $OC=\frac{1}{2} XC=\frac{y}{2}$ Therefore area of the trapezoid shaded area = $\frac{1}{2} \times (YC+AO) \times OC$= $\frac{1}{2} \times ( \frac{x}{2} + \frac{x}{4} ) \times \frac{y}{2}$ =$\frac{3xy}{16}=3$ (as $xy$=16) Categories ## Problem related to Money | AMC 8, 2002 | Problem 25 Try this beautiful problem from AMC-8, 2002 related to money (problem 25). Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group’s money does Ott now have? • $\frac{1}{3}$ • $\frac{1}{4}$ • $\frac{3}{4}$ ### Key Concepts Algebra Number theory fraction Answer:$\frac{1}{4}$ AMC-8 (2002) Problem 25 Challenges and Thrills in Pre College Mathematics ## Try with Hints Each Friend gave Ott the equal amount of money Can you now finish the problem ………. Assume that ott gets y dollars from each friend Can you finish the problem…….. Given that Ott gets equal amounts of money from each friend, we can say that he gets y dollars from each friend. This means that Moe has 5y dollars, Loki has 4y dollars, and Nick has 3y dollars. The total amount is 12y dollars, Therefore Ott gets 3y dollars total, Required fraction =$\frac{3y}{12y} = \frac{1}{4}$ Categories ## Problem from Probability | AMC 8, 2004 | Problem no. 21 Try this beautiful problem from Probability from AMC 8, 2004. ## Problem from Probability | AMC-8, 2004 | Problem 21 Spinners A and B  are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners’ numbers is even? • $\frac{2}{3}$ • $\frac{1}{3}$ • $\frac{1}{4}$ ### Key Concepts probability Equilly likely Number counting Answer: $\frac{2}{3}$ AMC-8, 2004 problem 21 Challenges and Thrills in Pre College Mathematics ## Try with Hints Even number comes from multiplying an even and even, even and odd, or odd and even Can you now finish the problem ………. A odd number only comes from multiplying an odd and odd………….. can you finish the problem…….. We know that even number comes from multiplying an even and even, even and odd, or odd and even and also a odd number only comes from multiplying an odd and odd, There are few cases to find the probability of spinning two odd numbers from  1 Multiply the independent probabilities of each spinner getting an odd number together and subtract it from  1 we get……. $1 – \frac{2}{4} \times \frac{2}{3}$= $1 – \frac{1}{3} = \frac{2}{3}$ Â Categories ## Intersection of two Squares | AMC 8, 2004 | Problem 25 Try this beautiful problem from Geometry based on Intersection of two Squares. ## When 2 Squares intersect | AMC-8, 2004 | Problem 25 Two $4\times 4$ squares intersect at right angles, bisecting their intersecting sides, as shown. The circle’s diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares? • $28-2\pi$ • $25-2\pi$ • $30-2\pi$ ### Key Concepts Geometry square Circle Answer: $28-2\pi$ AMC-8, 2004 problem 25 Pre College Mathematics ## Try with Hints Area of the square is $\pi (r)^2$,where $r$=radius of the circle Can you now finish the problem ………. Clearly, if 2 squares intersect, it would be a smaller square with half the side length, 2. can you finish the problem…….. Clearly the intersection of 2 squares would be a smaller square with half the side length, 2. The area of this region =Total area of larger two squares – the area of the intersection, the smaller square i.e $4^2+4^2 -2^2=28$ Now The diagonal of this smaller square created by connecting the two points of intersection of the squares is the diameter of the circle Using the Pythagorean th. diameter of the circle be $\sqrt{2^2 +2^2}=2\sqrt 2$ Radius=$\sqrt 2$ area of the square=$\pi (\sqrt2)^2$=$2\pi$ Area of the shaded region= 28-2$\pi$ Categories ## Probability | AMC 8, 2004 | Problem no. 22 Try this beautiful problem from Probability .You may use sequential hints to solve the problem. ## Probability | AMC-8, 2004 |Problem 22 At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is$\frac{2}{5}$. What fraction of the people in the room are married men? • $\frac{3}{8}$ • $\frac{1}{2}$ • $\frac{1}{4}$ ### Key Concepts probability combination Number counting Answer: $\frac{3}{8}$ AMC-8, 2007 problem 24 Challenges and Thrills in Pre College Mathematics ## Try with Hints Find the married men in the room … Can you now finish the problem ………. Find the total people can you finish the problem…….. Assume that there are 10 women in the room, of which $10 \times \frac{2}{5}$=4 are single and 10-4=6 are married. Each married woman came with her husband, so there are 6 married men in the room Total man=10+6=16 people Now The fraction of the people that are married men is $\frac{6}{16}=\frac{3}{8}$ Categories ## Area of Rectangle Problem | AMC 8, 2004 | Problem 24 Try this beautiful problem from Geometry from AMC-8, 2004 ,Problem-24, based on area of Rectangle. ## Rectangle | AMC-8, 2004 | Problem 24 In the figure ABCD is a rectangle and EFGH  is a parallelogram. Using the measurements given in the figure, what is the length  d  of the segment that is perpendicular to  HE and FG? • $7.1$ • $7.6$ • $7.8$ ### Key Concepts Geometry Rectangle Parallelogram Answer:$7.6$ AMC-8, 2004 problem 24 Pre College Mathematics ## Try with Hints Find Area of the Rectangle and area of the Triangles i.e $(\triangle AHE ,\triangle EBF , \triangle FCG , \triangle DHG)$ Can you now finish the problem ………. Area of the Parallelogram EFGH=Area Of Rectangle ABCD-Area of$(\triangle AHE +\triangle EBF + \triangle FCG + \triangle DHG)$ can you finish the problem…….. Area of the Rectangle =$CD \times AD$=$10 \times 8$=80 sq.unit Area of the $\triangle AHE$ =$\frac{1}{2} \times AH \times AE$= $\frac{1}{2} \times 4 \times 3$ =6 sq.unit Area of the $\triangle EBF$ =$\frac{1}{2} \times EB \times BE$= $\frac{1}{2} \times 6 \times 5$ =15 sq.unit Area of the $\triangle FCG$ =$\frac{1}{2} \times GC \times FC$= $\frac{1}{2} \times 4\times 3$ =6 sq.unit Area of the $\triangle DHG$ =$\frac{1}{2} \times DG \times DH$= $\frac{1}{2} \times 6 \times 5$ =15 sq.unit Area of the Parallelogram EFGH=Area Of Rectangle ABCD-Area of$(\triangle AHE +\triangle EBF + \triangle FCG + \triangle DHG)$=$80-(6+15+6+15)=80-42=38$ As ABCD is a Rectangle ,$\triangle GCF$ is a Right-angle triangle, Therefore GF=$\sqrt{4^2 + 3^2}$=5 sq.unit Now Area of the parallelogram EFGH=$GF \times d$=38 $\Rightarrow 5 \times d$=38 $\Rightarrow d=7.6$ Categories ## Radius of the Circle | AMC-8, 2005 | Problem 25 Try this beautiful problem from Geometry: Radius of a circle ## Radius of a circle – AMC-8, 2005- Problem 25 A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle? • $\frac{5}{\sqrt \pi}$ • $\frac{2}{\sqrt \pi}$ • $\sqrt \pi$ ### Key Concepts Geometry Cube square Answer: $\frac{2}{\sqrt \pi}$ AMC-8 (2005) Problem 25 Pre College Mathematics ## Try with Hints The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square Can you now finish the problem ………. Region within the circle and square be $x$ i.e In other words, it is the area inside the circle and the square The area of the circle -x=Area of the square – x can you finish the problem…….. Given that The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square Let the region within the circle and square be $x$ i.e In other words, it is the area inside the circle and the square . Let r be the radius of the circle Therefore, The area of the circle -x=Area of the square – x so, $\pi r^2 – x=4-x$ $\Rightarrow \pi r^2=4$ $\Rightarrow r^2 = \frac{4}{\pi}$ $\Rightarrow r=\frac{2}{\sqrt \pi}$

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Why is it important to learn to tie knots? There are a few people in each generation that just get a kick out of seeing how many of the approximately 4,000 different knots they can learn to tie. For most people knots keep your shoes from falling off, tie up bundles of limbs and twigs for trash pickup, fly a kite or tie a fly on a fishing line and all those other things that involve string, cord, line or rope. Each of the 4,000 or so knots has an application for which it is best but we really don't expect Cub Leaders to teach all 4,000 knots to the Cubs. The Boy Scouts have selected five knots that will fill most of the Cub's needs. These knots are Square Knot, Sheet Bend, Bowline, Two Half Hitches and Slipknot. Instructions for tying these knots are in the Bear Cub Scout book, The Scouting Book of Knots and the Boy Scout handbook as well as numerous non Boy Scout publications. The uses of these basic knots are: Square Knot - tie two ropes of the same size and material together. Sheet Bend - tie two ropes of different sizes and/or materials together. Bowline - tie a fixed non slipping loop in a rope. Two Half Hitches - all purpose hitch for attaching rope to an uneven shape. Slip Knot - can be used in place of two half hitches where it can be slipped over the end of an object to which it is to be attached. Try to teach Cub Scout's basic knots over an extended period. Do not try to teach them more than two knots in any given session. Most boys will start to get confused after the second knot. Repetition is the key to teaching knots. Once you have introduced them, keep including the skills in various games and other activities. Using rope of two different colors helps some boys see more clearly how knots are correctly tied and lessens confusion. Boys will need your individual attention in learning and demonstrating these knots, so try to get some help in watching them tie them. Your Den Chief would be good at this. You might ask him to bring a fellow Boy Scout along who has earned the Pioneering merit badge to help out when you are teaching the boys knots. The ends or a rope should be whipped or taped to hold the fibers in place. Instructions for this are in the Bear book. The rope should be stored dry. A natural fiber rope will rot if put away wet and manufactured fiber will mildew. The rope should be clean. Dirt in the rope will damage the fibers and weaken the rope. Remove all knots and kinks. Knots or kinks in a rope for a long period of time will damage the fibers and weaken the rope. Coil the rope as described in the Bear book. Games And Projects As a den project, you could have each boy make a small knotboard. Cut a board from plywood about 18 inches square. Staple knots to the board. Use dowel rods for hitches over bars. As the boy completes and passes each knot requirement, have him tie the knot on the board. When he is finished, you can have him hang his board where you meet as a den or he can take it home. The knot board will serve as a token of accomplishment as well as a reminder to the boy of how these knots are tied. Friendship Circle Closing Each den member is given a three-foot length of rope which he ties to his neighbor's with a square knot so that a circle is made. Boys pull back on the line with their left hands and make the Cub Scout sign with their right. Den leader says, "This circle shows the bond of friendship we have in Cub Scouting. Now please join me in the Cub Scout Promise." Save My Child Divide the den into two teams. One boy on each team is the child. He sits down on several sheets of newspaper about 15 feet from his teammates. Each of the others has a three-foot length of rope. On signal, the first boy on each team ties a bowline with a small loop in his rope and hands the other end to the next boy. He ties on his rope with a square knot. In turn, all others attach their ropes with square knots. When all knots are tied, the team leader casts the rope to the child, who grasps it by the bowline loop. Then the child holds the newspapers with his free hand and is pulled in by his team. First team finished wins, provided that all knots are correct. Knot Step Contest Line up Cubs at one end of the room. Each is given a 6 foot length of rope. Call out the name of a knot. Each Cub ties the knot. Judges quickly check the knots. Each Cub who tied the knot correctly can take one step forward. The process is repeated until the first Cub (winner) reaches a predetermined mark.

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The crossbow was powerful in battle because its bolts (a shorter version of arrows) could penetrate the chain mail of opponents. Leaders of the Catholic Church, outraged by the lethal weapon, deemed the crossbow "unfit to be used among Christians" and tried to ban it from warfare in 1139without success. Richard I ("the Lionheart") died of a crossbow wound on the battlefield in 1199. English troops eventually came to favor the longbow in warfare, but the crossbow, which was much easier to use, remained popular in Italy and several other European countries. Under the feudal system of the Middle Ages, when landowners were responsible for their own local defense, the crossbow was prized by town militia and troops of mercenaries. "The protection of one's castle or town was important. Civil outbreaks and invasions happened often. Perhaps it was more commonplace in Italy because there were so many city-states," said Pyhrr. "Many commercial towns had their own militia, and groups of crossbowmen were part of the militia." A number of Italian towns, he noted, became famous for their skilled crossbowmen. "It seems to have become something of a major export, a commoditymercenaries that could be rented out. It brought fame to Genoa and some other cities," said Pyhrr. Hunting and Sport The crossbow became obsolete in warfare in the 15th century after firearms were introduced. It continued to be used widely for hunting, however, and also became popular for sport. "Shooting societies remained in place from the 16th century," said Pyhrr. Companies of crossbowmen organized inter-city competitions to display their skills and keep the ancient art alive. Crossbows are still used for hunting in some parts of the world, especially Africa and Southeast Asia, but the practice has been outlawed or restricted in many countries, including the United States. The basic design of the crossbow has changed little through the ages; it consists of a bow fixed horizontally atop a stock, or handle; string; nocks, or notched end pieces of the bow to which the string is attached; and a trigger to release the arrow-like bolts. The bolts are made of wood, with tips of steel and feather fins. Medieval crossbows had a bow constructed of composite layers, to make it "soft and springy," said Pyhrr. The materials for the various parts were all natural, including wood, bone, tendon, and glue. Steel-made bows, which were stronger than composite bows, came into use by the 14th to 15th century. Many modern crossbows are made of lighter materials and have sighting mechanisms and other modifications to make them more efficient. Today, the Gubbio and Sansepolcro contenders are required to use crossbows modeled on the traditional design, although the style and size can vary slightly to suit the individual owners and teams. "Some balestre (crossbows) are handed down from the families, but some of the balestrieri are very good at creating and producing them for competition," Ubaldo said. "To become a balestriere is very hard," Ubaldo explained. An aspiring bowman must practice and master the sport on his own, and pass an exam to demonstrate knowledge about crossbow engineering and the traditional palio. A committee of highly experienced marksmen in the Crossbowmen's Society ultimately decides whether to admit the newcomer into the group. "To enter is quite impossible. There have been just two new balestrieri in the last four years," said Ubaldo. Last May the crossbow competition in Gubbio was held in the plaza between the 14th-century Palace of the Consuls and the town hall. The activities featured 600-year-old music, colorful flag-throwing exhibitions, and men positioned precariously inside a high open tower to ring the town's massive bell. Officials and other representatives of the two competing towns accompanied the teams to the shooting grounds. Many of them were dressed in costumes inspired by figures in the famous Renaissance paintings of Piero della Francesca, a native son of Sansepolcro. Patti Absher, the owner of Great Travels in Washington, D.C., who has taken tour groups to the Gubbio event for the past five years, noted the striking resemblance of the characters in the paintings and their modern-day counterparts. "It's not just the costumes that are so amazingit's also the faces," she said. One after another, the 41 bowmen from Gubbio and 46 from Sansepolcro took their places at the cobbler-style shooting benches. In rapid successionthwack, thwack, thwackmost of the bolts hit the black-and-white target 118 feet (36 meters) away. As the bolts piled up in a massive cluster, some were displaced by others and fell to the ground. Later, when the bull's-eye was removed and examined closely by a committee, this year's winner was Marcello Pasquinia sixtyish dark-horse contender. "Nobody expected that from him. He is really an outsider. His turn to throw the arrow is around the 40th to 50th place," said Ubaldo. With Antonio Madonnini and Claudio Mancini placing second and third, the event was a sweep for the Gubbio team. Bearing the coveted palio banner, made this year by Florence artist Imperio Nigiani, the joyous Pasquini was hoisted onto the shoulders of his teammates and carried through the streets of Gubbio. For this story, Paolo Clementi of Gubbio did the English translation of quotes and information provided by Orlandi Ubaldo. SOURCES AND RELATED WEB SITES

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# Maximum Subarray OR Largest Sum Contiguous Subarray Problem – Divide and Conquer Objec­tive:  The max­i­mum sub­ar­ray prob­lem is the task of find­ing the con­tigu­ous sub­ar­ray within a one-dimensional array of num­bers which has the largest sum. Exam­ple: ```int [] A = {−2, 1, −3, 4, −1, 2, 1, −5, 4}; Output: contiguous subarray with the largest sum is 4, −1, 2, 1, with sum 6. ``` Approach: Ear­lier we have seen how to solve this prob­lem using Kadane’s Algo­rithm and using Dynamic pro­gram­ming. In this arti­cle we will solve the prob­lem using divide and conquer. 1.     Task is to find out sub­ar­ray which has the largest sum. So we need to find out the 2 indexes (left index and right index) between which the sum is maximum. 2.     Divide the prob­lem into 2 parts, left half and right half. 1. Now we can con­clude the final result will be in one of the three possibilities. 1. Left half of the array. (Left index and right index both are in left half of the array). 2. Right half of the array. (Left index and right index both are in right half of the array). 3. Result will cross the mid­dle of the array. (Left index in the and left half of the array and right index in right half of the array). 2. Solve the all three and return the max­i­mum among them. 3. Left half and right half of the array will be solved recursively. Com­plete Code: Out­put: ```Maximum Sub Array sum is : 17 ```

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In today’s world, technology plays a crucial role in shaping the future. Young professionals seeking to build their careers in the fields of science, technology, engineering, and mathematics (STEM) must be equipped with the latest tools and techniques. One such tool that has gained popularity in recent years is the 3D printer. With its ability to create complex designs and prototypes, a 3D printer has become an indispensable tool for young professionals in the world of STEM. The Snapmaker 3D printer is an example of a professional 3D printer that can be used to integrate into the world of STEM. The printer is designed for precision and accuracy, making it ideal for creating intricate and detailed models. Its high-speed printing capabilities allow users to create models quickly and efficiently, making it a valuable tool for young professionals who are looking to stay ahead of the curve. One way that a 3D printer can be integrated into the world of STEM for young professionals is through the creation of prototypes. A prototype is a preliminary model of a product that is used to test and refine its design. With a 3D printer, young professionals can create prototypes quickly and accurately, allowing them to test their designs in real-world scenarios. This can be particularly useful for engineers and designers who are working on complex projects, such as robotics or biomedical devices. Another way that a 3D printer online can be integrated into the world of STEM for young professionals is through the creation of custom parts. Often, off-the-shelf components are not suitable for specific projects, and custom parts need to be created. A 3D printer allows young professionals to create custom parts quickly and easily, without having to rely on expensive manufacturing processes. This can be particularly useful for those working in the fields of robotics, aerospace, or automotive engineering. In addition to creating prototypes and custom parts, a 3D printer online can also be used to create educational models. These models can be used to help students understand complex concepts in a visual and interactive way. For example, a 3D printer can be used to create a model of the human heart or the solar system, allowing students to explore these concepts in a more engaging way. This can be particularly useful for educators and researchers who are looking for new and innovative ways to teach STEM subjects. Finally, a 3D printer can be used to create art and design projects. This may seem unrelated to STEM at first glance, but creativity and innovation are crucial elements of STEM fields. A 3D printer allows young professionals to explore their creativity and design skills, creating unique and interesting projects that can be used in a variety of applications. For example, a designer could use a 3D printer to create a custom piece of non-metal ornaments or a sculpture. A 3D printer online can be integrated into the world of STEM for young professionals in a variety of ways. From creating prototypes and custom parts to designing educational models and exploring creativity and design, a professional 3D printer like the Snapmaker can be a valuable tool for young professionals looking to build their careers in STEM. With its precision, accuracy, and high-speed printing capabilities, a 3D printer can help young professionals stay ahead of the curve and bring their ideas to life in new and innovative ways. If you are a parent looking to get your kids to lean into STEM or you are looking for Montessori tools to enhance cognitive stuff, this is one of the best tools you can have around. A 3D printer from Snapmaker can help make these dreams and perhaps experiments come through.

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# 2.4 Thin lenses  (Page 5/13) Page 5 / 13 $\frac{{n}_{2}}{{d}_{\text{o}}^{\prime }}+\frac{{n}_{1}}{{d}_{\text{i}}}=\frac{{n}_{1}-{n}_{2}}{{R}_{2}}.$ The image is real and on the opposite side from the object, so ${d}_{\text{i}}>0$ and ${d}_{\text{o}}^{\prime }>0$ . The second surface is convex away from the object, so ${R}_{2}<0$ . [link] can be simplified by noting that ${d}_{\text{o}}^{\prime }=|{d}_{\text{i}}^{\prime }|+t$ , where we have taken the absolute value because ${d}_{\text{i}}^{\prime }$ is a negative number, whereas both ${d}_{\text{o}}^{\prime }$ and t are positive. We can dispense with the absolute value if we negate ${d}_{\text{i}}^{\prime }$ , which gives ${d}_{\text{o}}^{\prime }=\text{−}{d}_{\text{i}}^{\prime }+t$ . Inserting this into [link] gives $\frac{{n}_{2}}{-{d}_{\text{i}}^{\prime }+t}+\frac{{n}_{1}}{{d}_{\text{i}}}=\frac{{n}_{1}-{n}_{2}}{{R}_{2}}.$ $\frac{{n}_{1}}{{d}_{\text{o}}}+\frac{{n}_{1}}{{d}_{\text{i}}}+\frac{{n}_{2}}{{d}_{\text{i}}^{\prime }}+\frac{{n}_{2}}{-{d}_{\text{i}}^{\prime }+t}=\left({n}_{2}-{n}_{1}\right)\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right).$ In the thin-lens approximation    , we assume that the lens is very thin compared to the first image distance, or $t\ll {d}_{\text{i}}^{\prime }$ (or, equivalently, $t\ll {R}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{R}_{2}$ ). In this case, the third and fourth terms on the left-hand side of [link] cancel, leaving us with $\frac{{n}_{1}}{{d}_{\text{o}}}+\frac{{n}_{1}}{{d}_{\text{i}}}=\left({n}_{2}-{n}_{1}\right)\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right).$ Dividing by ${n}_{1}$ gives us finally $\frac{1}{{d}_{\text{o}}}+\frac{1}{{d}_{\text{i}}}=\left(\frac{{n}_{2}}{{n}_{1}}-1\right)\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right).$ The left-hand side looks suspiciously like the mirror equation that we derived above for spherical mirrors. As done for spherical mirrors, we can use ray tracing and geometry to show that, for a thin lens, $\frac{1}{{d}_{\text{o}}}+\frac{1}{{d}_{\text{i}}}=\frac{1}{f}$ where f is the focal length of the thin lens (this derivation is left as an exercise). This is the thin-lens equation. The focal length of a thin lens is the same to the left and to the right of the lens. Combining [link] and [link] gives $\frac{1}{f}=\left(\frac{{n}_{2}}{{n}_{1}}-1\right)\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right)$ which is called the lens maker’s equation . It shows that the focal length of a thin lens depends only of the radii of curvature and the index of refraction of the lens and that of the surrounding medium. For a lens in air, ${n}_{1}=1.0$ and ${n}_{2}\equiv n$ , so the lens maker’s equation reduces to $\frac{1}{f}=\left(n-1\right)\left(\frac{1}{{R}_{1}}-\frac{1}{{R}_{2}}\right).$ ## Sign conventions for lenses To properly use the thin-lens equation, the following sign conventions must be obeyed: 1. ${d}_{\text{i}}$ is positive if the image is on the side opposite the object (i.e., real image); otherwise, ${d}_{\text{i}}$ is negative (i.e., virtual image). 2. f is positive for a converging lens and negative for a diverging lens. 3. R is positive for a surface convex toward the object, and negative for a surface concave toward object. ## Magnification By using a finite-size object on the optical axis and ray tracing, you can show that the magnification m of an image is $m\equiv \frac{{h}_{\text{i}}}{{h}_{\text{o}}}=\text{−}\frac{{d}_{\text{i}}}{{d}_{\text{o}}}$ (where the three lines mean “is defined as”). This is exactly the same equation as we obtained for mirrors (see [link] ). If $m>0$ , then the image has the same vertical orientation as the object (called an “upright” image). If $m<0$ , then the image has the opposite vertical orientation as the object (called an “inverted” image). ## Using the thin-lens equation The thin-lens equation and the lens maker’s equation are broadly applicable to situations involving thin lenses. We explore many features of image formation in the following examples. Consider a thin converging lens. Where does the image form and what type of image is formed as the object approaches the lens from infinity? This may be seen by using the thin-lens equation for a given focal length to plot the image distance as a function of object distance. In other words, we plot ${d}_{\text{i}}={\left(\frac{1}{f}-\frac{1}{{d}_{\text{o}}}\right)}^{-1}$ for a given value of f . For $f=1\phantom{\rule{0.2em}{0ex}}\text{cm}$ , the result is shown in part (a) of [link] . The photoelectric effect is the emission of electrons when light shines on a material. What is photoelectric effect it gives practical evidence of particke nature of light. Omsai particle nature Omsai photoelectric effect is the phenomenon of emission of electrons from a material(i.e Metal) when it is exposed to sunlight. Emitted electrons are called as photo electrons. Anil what are the applications of quantum mechanics to medicine? Neptune application of quantum mechanics in medicine: 1) improved disease screening and treatment ; using a relatively new method known as BIO- BARCODE ASSAY we can detect disease-specific clues in our blood using gold nanoparticles. 2) in Genomic medicine 3) in protein folding 4) in radio theraphy(MRI) Anil Quantam physics ki basic concepts? why does not electron exits in nucleaus electrons have negative YASH Proton and meltdown has greater mass than electron. So it naturally electron will move around nucleus such as gases surrounded earth Amalesh .......proton and neutron.... Amalesh excuse me yash what negative Rika coz, electron contained minus ion Manish negative sign rika shrestha ji YASH electron is the smallest negetive charge...An anaion i.e., negetive ion contains extra electrons. How ever an atom is neutral so it must contains proton and electron Amalesh yes yash ji Rika yes friends Prema koantam theory Laxmikanta yes prema Rika quantum theory tells us that both light and matter consists of tiny particles which have wave like propertise associated with them. Prema proton and nutron nuclear power is best than proton and electron kulamb force Laxmikanta what is de-broglie wave length? Ramsuphal plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it. expression for photon as wave Are beta particle and eletron are same? yes mari how can you confirm? Amalesh sry Saiaung If they are same then why they named differently? Amalesh because beta particles give the information that the electron is ejected from the nucleus with very high energy Absar what is meant by Z in nuclear physic atomic n.o Gyanendra no of atoms present in nucleus Sanjana Note on spherical mirrors what is Draic equation? with explanation what is CHEMISTRY it's a subject Akhter it's a branch in science which deals with the properties,uses and composition of matter Eniabire what is a Higgs Boson please? god particles is know as higgs boson, when two proton are reacted than a particles came out which is used to make a bond between than materials M.D bro little abit getting confuse if i am wrong than please clarify me M.D the law of refraction of direct current lines at the boundary between two conducting media of what is the black body of an ideal radiator uncertainty principles is applicable to Areej fermions FRANKLINE what is the cause of the expanding universe? FRANKLINE microscopic particles or gases Areej Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart. One explanation for dark energy is that it is a property of space. Areej FRANKLINE no problem Areej what is photoelectric equation

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# Video: GCSE Mathematics Foundation Tier Pack 5 β€’ Paper 2 β€’ Question 2 GCSE Mathematics Foundation Tier Pack 5 β€’ Paper 2 β€’ Question 2 01:37 ### Video Transcript Simplify four 𝑦 minus two 𝑦 plus 𝑦. To simplify an algebraic expression like the one we have here means that we need to collect like terms. Like terms are the terms which have the same letters and powers. For example, two 𝑦 and three 𝑦 squared are not like terms as although they have the same letters, they have different powers. The first term just has 𝑦, which is 𝑦 to the power of one, and the second term has 𝑦 squared. In our expression, all of the terms are actually like terms as they all have the same letter 𝑦 and they all have the same power, which although it isn’t written, is a power of one. To simplify then, we just need to look at how many 𝑦s we have and how many we’re adding or subtracting. The first part of the expression is four 𝑦 minus two 𝑦. And if you have four of something and then subtract two of the same thing, you’re left with two of them. So four 𝑦 minus two 𝑦 simplifies to two 𝑦. And therefore, the whole expression simplifies to two 𝑦 plus 𝑦. Now, remember 𝑦 is just a shorthand way of writing one 𝑦. We never write one 𝑦 in algebra. We just write it as 𝑦. So this means two 𝑦 plus one 𝑦. And as two plus one is equal to three, this simplifies to three 𝑦. The expression four 𝑦 minus two 𝑦 plus 𝑦 simplifies to three 𝑦.

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# A few questions involving square roots and inequalities • Sep 6th 2011, 11:08 PM juliak A few questions involving square roots and inequalities • Sep 6th 2011, 11:49 PM Prove It re: A few questions involving square roots and inequalities For the first one, $\displaystyle \sqrt{x - 1}$ is not equal to $\displaystyle \sqrt{x} - 1$. You should start by squaring both sides. • Sep 7th 2011, 12:04 AM Prove It re: A few questions involving square roots and inequalities Quote: Originally Posted by juliak For the last one, you first need to note that $\displaystyle x \neq 3$ (Why?) Then \displaystyle \begin{align*} \frac{\sqrt{x^3 - 6x^2 + 9x}}{x-3} &= \frac{\sqrt{x(x^2 - 6x+ 9)}}{x - 3} \\ &= \frac{\sqrt{x}\sqrt{x^2 - 6x + 9}}{x - 3} \\ &= \frac{\sqrt{x}\sqrt{(x - 3)^2}}{x-3} \textrm{ which means }x \geq 0 \\ &= \frac{\sqrt{x}|x-3|}{x-3} \\ &= \begin{cases}\frac{\sqrt{x}(x - 3)}{x-3}\textrm{ if }x - 3 > 0 \\ \frac{\sqrt{x}\,\left[-(x-3)\right]}{x-3}\textrm{ if }x - 3 < 0\end{cases} \\ &= \begin{cases}\sqrt{x}\textrm{ if }x > 3 \\ -\sqrt{x} \textrm{ if }0 \leq x < 3 \textrm{ since we have already established that }x \geq 0\end{cases}\end{align*} You should be able to sketch this from here. • Sep 7th 2011, 12:20 AM Prove It re: A few questions involving square roots and inequalities Quote: Originally Posted by juliak b) Your working has been marked incorrect because you have failed to realise that multiplying or dividing both sides of an inequality changes the inequality sign. It's quite possible that your denominators could be negative. The easiest way to do this is first to note that $\displaystyle x \neq -1$ and $\displaystyle x \neq 1$, then... \displaystyle \begin{align*} \frac{1}{1 + x} &< \frac{x}{x - 1} \\ \frac{1}{1 + x} &< \frac{x - 1 + 1}{x - 1} \\ \frac{1}{1 + x} &< \frac{x - 1}{x - 1} + \frac{1}{x - 1} \\ \frac{1}{1 + x} &< 1 + \frac{1}{x - 1} \\ \frac{1}{1 + x} - \frac{1}{x - 1} &< 1 \\ \frac{(x - 1) - (1 + x)}{(1 + x)(x - 1)} &< 1 \\ \frac{-2}{(1 + x)(x - 1)} &< 1 \end{align*} Now to solve this, you need to consider two cases, the first where $\displaystyle (1 + x)(x - 1) < 0$ and the second where $\displaystyle (1 + x)(x - 1) > 0$. Case 1: $\displaystyle (1 + x)(x - 1) < 0 \implies x^2 - 1 < 0 \implies |x| < 1 \implies -1 < x < 1$ which gives \displaystyle \begin{align*} \frac{-2}{(1 + x)(x - 1)} &< 1 \\ -2 &> (1 + x)(x - 1) \\ -2 &> x^2 - 1 \\ x^2 - 1 &< -2 \\ x^2 &< -1 \textrm{ which is not true for any }x \end{align*} Case 2: $\displaystyle (1 + x)(x - 1) > 0 \implies x^2 - 1 > 0 \implies |x| > 1 \implies x < -1 \textrm{ or }x > 1$ which gives \displaystyle \begin{align*} \frac{-2}{(1 + x)(x - 1)} &< 1 \\ -2 &< (1 + x)(x - 1) \\ -2 &< x^2 - 1 \\ x^2 - 1 &> -2 \\ x^2 &> -1\textrm{ which is true for all possible }x \end{align*} So the solution is $\displaystyle x < -1 \textrm{ and }x > 1$. • Sep 7th 2011, 12:34 AM Prove It re: A few questions involving square roots and inequalities Quote: Originally Posted by juliak \displaystyle \begin{align*} |4x - x^3| &= \begin{cases}4x - x^3 \textrm{ if }4x - x^3 \geq 0 \\ x^3 - 4x \textrm{ if }4x - x^4 < 0\end{cases} \end{align*} So to work out the possible $\displaystyle x$ values for each case, we need to solve the equation $\displaystyle f(x) = 4x - x^3 = 0$, since the function changes sign at these solutions. \displaystyle \begin{align*} 4x - x^3 &= 0 \\ x(4 - x^2) &= 0 \\ x(2 - x)(2 + x) &= 0 \\ x = -2 \textrm{ or }x &= 0 \textrm{ or }x = 2 \end{align*} Now testing values for the function... \displaystyle \begin{align*} f(-3) &= 4(-3) - (-3)^3 \\ &= -12 - (-27) \\ &= 27 - 12 \\ &= 15 \end{align*} so $\displaystyle 4x - x^3 > 0\textrm{ if } x < -2$. \displaystyle \begin{align*}f(-1) &= 4(-1) - (-1)^3 \\ &= -4 - (-1) \\ &= 1 - 4 \\ &= -3\end{align*} so $\displaystyle 4x - x^3 < 0 \textrm{ if }-2 < x < 0$. \displaystyle \begin{align*} f(1) &= 4(1) - 1^3 \\ &= 4 - 1 \\ &= 3 \end{align*} so $\displaystyle 4x - x^3 > 0 \textrm{ if } 0 < x < 2$. \displaystyle \begin{align*} f(3) &= 4(3) - 3^3 \\ &= 12 - 27 \\ &= -15 \end{align*} so $\displaystyle 4x - x^3 < 0 \textrm{ if } x > 2$. Therefore $\displaystyle |4x - x^3| = \begin{cases} 4x - x^3 \textrm{ if }x \leq 2 \textrm{ or }0 \leq x \leq 2 \\ x^3 - 4x \textrm{ if }-2 < x < 0 \textrm{ or }x > 2 \end{cases}$ • Sep 7th 2011, 07:13 AM mathjeet Re: A few questions involving square roots and inequalities $\sqrt(x - 1) = \sqrt(1-x)$ is true only if x = 1 which is the solution.

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Welcome Zoom Room Jamboard Textbook Lectures Drive Calculator Skill Topics Application Topics Resources # Math OERWeek 9 Homework, Part A The greatest reward for a student is not a good grade. It is the willingness of his teacher to listen to him. - Nikolay Konstantinov Answer every question. Try being nice to your eyes and posture by printing this page and working with pencil and paper. Then use the button at the bottom of the page to create a code by processing your answers. Copy-and-paste the code into an e-mail along with the answer to your short answer question. Keep trying each homework assignment until you get 8 out of 10 or more. 1. At a restaurant, you can choose from 3 appetizers, 11 entrees, and 4 desserts. How many different three-course meals can you have? 3 × 11 × 4 3! × 11! × 4! (3 × 11 × 4)! 3 + 11 + 4 3! + 11! + 4! (3 + 11 + 4)! 2. A pianist wants to play five pieces at a recital. In how many orders can the pianist arrange these pieces in the program? 5 25 120 125 3. In how many ways can first, second, and third prizes be awarded in a contest with 175 contestants? 525 1,050 877,975 5,267,850 175! 4. A local pizza shop is having a Five Topping Special. In how many ways can 5 pizza toppings be chosen from 14 available toppings? 70 120 1,680 2,002 240,240 5. The serial number on a dollar bill consists of a letter, followed by eight digits and then a letter. How many different serial numbers are possible if the letters and digits cannot be repeated? between 1 billion and 2 billion less than 1 billion between 2 billion and 3 billion more than 3 billion 6. The serial number on a dollar bill consists of a letter, followed by eight digits and then a letter. How many different serial numbers are possible if the letters and digits can be repeated? between 1 billion and 2 billion less than 1 billion between 2 billion and 3 billion more than 3 billion 7. Two kids play a game involving flipping two coins and keeping score. In each round, they flip the coins one at a time. One kid gets a point whenever the first coin is heads, and the other kid gets a point whenever the second coin is heads. They are surprised when three consecutive rounds happen without any points. What is the chance that this happens? 0.8% 1.6% 2.4% 4.2% 8.3% 25% 8. With a shuffled normal deck of playing cards, what is the probability of a randomly chosen card being either a red card or less than (but not equal to) five? Consider aces to be less than five. 58% 65% 81% 85% 9. A hospital needs to hire people to be on-call to provide language interpretation services. The Venn diagram shows the languages requested last month. (An overlap shows where a patient could manage with either Italian or Spanish, for example). What is the probability that one interpreter who speaks Spanish and Chinese will be sufficient for the next interpretation request? 46% 53% 57% 65% 10. A group of atheletes consists of 12 people: 8 men and 4 women. What is the probability that a randomly selected team of 3 people would be all of one gender? 0.2% 0.5% 24% 27%

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## How do you write a trip essay? How To Write a Good Travel EssaySelect Your Favorite City. Sometimes a trip is explicitly taken to collect information for an essay. Choose a Few Attractions. Write a Compelling First Paragraph. Show Rather Than Tell. Use Images. Keep It Simple. Describe What You Achieved. Give Readers a Good Ending. How do you name an angle in three ways? The best way to describe an angle is with three points. One point on each ray and the vertex always in the middle. That angle could be NAMED in three ways: X, BXC, or CXB. Adjacent angles are two angles that have a common vertex, a common side, and no common interior points. What are 4 ways to name an angle? acute angle-an angle between 0 and 90 degrees. right angle-an 90 degree angle. obtuse angle-an angle between 90 and 180 degrees. straight angle-a 180 degree angle. ### What are the names of three collinear points? What are the names of three collinear points? Points L, J, and K are collinear. What are the sides of an angle? The sides of an angle refer to the two rays or line segments that form the angle. In the figure below, rays BA and BC are the sides of angle ABC. An angle is formed by rotating a ray around its endpoint. What do you call the side of a triangle? A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. ## How do I know my SSS SAS ASA AAS? There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.SSS (side, side, side) SSS stands for “side, side, side” and means that we have two triangles with all three sides equal. SAS (side, angle, side) ASA (angle, side, angle) AAS (angle, angle, side) HL (hypotenuse, leg) How do you find the sides of a triangle with 3 angles? In this case, use The Law of Sines first to find either one of the other two angles, then use Angles of a Triangle to find the third angle, then The Law of Sines again to find the final side.

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A: This is what we call a journey-based question, wherein an object is travelling from one starting point to an ending point. Here, a rollercoaster carriage is moving up from the bottom of the slope, and stopping at the top of the slope. To solve such journey-based questions, we’ll use the Principle of Conservation of energy to our advantage: since energy cannot be created nor can it be destroyed, it can only transform from one form to another, such that the total energy remains constant. At the bottom of the slope, the carriage only has kinetic energy because it is in motion. At the top of the slope, it has only gravitational potential energy. Note that due to negligible drag, there is no work done against frictional or resistive forces. Since total energy is constant throughout, energy at the start (at A) equals energy at the end (at B): Now, we can use trigonometry to find h:

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Cellular respiration is a metabolic process involving many different reactions taking place; organic compounds are oxidized in these reactions and energy is produced. These organic compounds are known as respiratory substrates. Respiratory substrates include: When these substrates are oxidized cells can use the energy produced to carry out their normal processes. The main aim of cell respiration is to produce energy; this energy is in the form of ATP. The many reactions which take place during cellular respiration can be divided into three stages: - Glycolysis: This stage involves the oxidation of glucose forming pyruvate and takes place in the cytoplasm of the cell (2ATP’s produced) - Krebs cycle: Breaks down pyruvate from glycolysis forming carbon dioxide and hydrogen (in the presence of oxygen) or forms ethanol/lactate (in the absence of oxygen). This stage takes place in the matrix of the mitochondria. (2ATP’s produced) - Electron Transfer System: In this stage the hydrogen is oxidized by oxygen forming water and takes place in the inner membrane of the mitochondria. (34ATP’s produced) |Cellular Respiration Image| Somewhere along these stages numerous amounts of ATP are produced which can then be utilized by the cell itself. The availability of oxygen is what determines the fate of pyruvate, therefore different products can be obtained from these stages depending on whether oxygen is made available or not. There are two types of respiration, these are: - Aerobic Respiration - Anaerobic Respiration When glucose is broken down in the cytoplasm the product formed is known as pyruvate. From this point the pyruvate can take one of two routes depending on whether oxygen is available or not. In aerobic respiration oxygen is made available. The purpose of this oxygen is to oxidize the pyruvate from glycolysis to ultimately form carbon dioxide and water (ATP’s are also produced along the way). The first stage involves the Krebs cycle where the pyruvate was broken to form carbon dioxide and hydrogen. This hydrogen then moves along to hydrogen carriers in the second stage (electron transfer chain) where it goes through a number of carriers and finally gets oxidized by oxygen forming water. Combining the equations from glycolysis, Krebs cycle and the electron transfer chain we get: C6H12O6 + 6O2 => 6CO2 +6H2O + 38ATP (Energy) Note that 38 ATP’s are produced in aerobic respiration, compare this with the amount of ATP’s produced during anaerobic respiration and you should see why aerobic respiration is more efficient than anaerobic respiration. Anaerobic Respiration (Fermentation) In this type of respiration oxygen is absent which therefore means different products will be formed from the point where pyruvate left glycolysis. It might seem odd but there are organisms that can actually survive with and without oxygen, such organisms are termed facultative anaerobes. There are even organisms that can’t survive in the presence of oxygen; these organisms are called obligate anaerobes. Remember from aerobic respiration that the purpose of the oxygen was to combine with hydrogen in the final stage to ultimately form water. However since oxygen is not present there is no acceptor at the end of the final stage (electron transfer chain) to combine with hydrogen. Since there is no final acceptor the hydrogen goes back and combines with the pyruvate preventing the release of any energy. Anaerobic Respiration has two Pathways which depend on the type of organism: 1. Anaerobic Respiration in Fungi In fungi such as yeast Pyruvate is ultimately converted into ethanol and carbon dioxide (Alcoholic Fermentation) with an overall production of 2 ATP’s, allot of energy however still remains locked within ethanol. C6H12O6 => 2Ethanol + 2CO2 + 2ATP (Energy) Despite being an inefficient source of energy production this process still has some uses. Uses of fermentation by Yeast include: - The production of alcoholic drinks such as wine and beer. - The Carbon dioxide produced is used in the manufacture of bread, because it allows for the dough to rise. - Though the energy still remains trapped in ethanol in countries such as Brazil it is used to make gasohol which can then be used to fuel cars. 2. Anaerobic Respiration in Animals In certain tissues such as muscle tissues in animals the pyruvate formed from glycolysis is ultimately converted to lactate with the formation of 2ATP’s, no carbon dioxide is produced in this type of anaerobic respiration. In animals buildup of lactate can result in a sensation of fatigue and cramps. C6H12O6 => 2Lactate + 2ATP (Energy) As is the case with ethanol in fermentation much of the energy remains locked within the lactate which therefore means this path is also inefficient. The energy can however be released from the lactate if oxygen is later made available. |Image Showing Aerobic and Anaerobic Respiration|

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## Conversion formula The conversion factor from hours to weeks is 0.005952380952381, which means that 1 hour is equal to 0.005952380952381 weeks: 1 hr = 0.005952380952381 wk To convert 32.1 hours into weeks we have to multiply 32.1 by the conversion factor in order to get the time amount from hours to weeks. We can also form a simple proportion to calculate the result: 1 hr → 0.005952380952381 wk 32.1 hr → T(wk) Solve the above proportion to obtain the time T in weeks: T(wk) = 32.1 hr × 0.005952380952381 wk T(wk) = 0.19107142857143 wk The final result is: 32.1 hr → 0.19107142857143 wk We conclude that 32.1 hours is equivalent to 0.19107142857143 weeks: 32.1 hours = 0.19107142857143 weeks ## Alternative conversion We can also convert by utilizing the inverse value of the conversion factor. In this case 1 week is equal to 5.2336448598131 × 32.1 hours. Another way is saying that 32.1 hours is equal to 1 ÷ 5.2336448598131 weeks. ## Approximate result For practical purposes we can round our final result to an approximate numerical value. We can say that thirty-two point one hours is approximately zero point one nine one weeks: 32.1 hr ≅ 0.191 wk An alternative is also that one week is approximately five point two three four times thirty-two point one hours. ## Conversion table ### hours to weeks chart For quick reference purposes, below is the conversion table you can use to convert from hours to weeks hours (hr) weeks (wk) 33.1 hours 0.197 weeks 34.1 hours 0.203 weeks 35.1 hours 0.209 weeks 36.1 hours 0.215 weeks 37.1 hours 0.221 weeks 38.1 hours 0.227 weeks 39.1 hours 0.233 weeks 40.1 hours 0.239 weeks 41.1 hours 0.245 weeks 42.1 hours 0.251 weeks

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Context and definition Within the framework of the Deepwater Horizon incident, the aborted attempt to install a containment chamber on the main leak in May 2010 was due to the formation of methane hydrates crystals. They appear as ice-like crystals. Methane molecules are in fact trapped in “cages” comprised of water molecules. This type of structure can only be seen in areas of high pressure and/or low temperature. Observations suggest that in the open environment, hydrates do not form visible structures in a long-lasting way. It is only when they come up against an obstacle that they form crystals. There are two main types of natural methane hydrate deposits: - oceanic deposits which may be located in the depths of marine sediments or on continental slopes - continental deposits in Arctic permafrost. Risks related to extraction Natural deposits of methane hydrates are a potential energy source capable of satisfying the world’s energy requirements for thousands of years. However, extracting this substance generates environmental and safety-related problems. First, using more methane would mean releasing even more greenhouse gases into the atmosphere. Secondly, trapped methane hydrates must be handled extremely carefully as if the gas is suddenly discharged, there is a risk of massive release.

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# Homework Help: HELP: Fourier series Problem 1. May 21, 2006 ### Hummingbird25 Hi Given the function $$f(t) = t^2$$, were $$t \in ]- \pi, pi[$$, and is continious find the fourier series for f(t). $$L = 2 \pi$$. Then $$A_0 = \frac{1}{2 \pi} \int \limit_{-\pi} ^{\pi} t^2 dt = \frac{\pi ^2}{3}$$ $$A_n = \int \limit_{-\pi} ^{\pi} t^2 \cdot cos(\matrm{n} \pi \mathrm{t}) dt$$ $$A_n = \int \limit_{-\pi} ^{\pi} u^2 \cdot cos(u) du$$, where u = n \pi t, The new limit gives: $$A_n = \int \limit_{0} ^{2 \pi^2 n} u^2 \cdot cos(u) du$$ $$A_n = [u^2 \cdot sin(u) - 2sin(u) + 2u sin(u)]_{0} ^{2 \pi ^ 2 n}$$ Then $$B_n = \int \limit_{0} ^{2n \pi^2} u^2 \cdot sin(u) du$$ which gives $$B_n = (2-4n^2 * \pi ^4 * cos(2n * \pi ^2) + 4n *sin (2n * \pi ^2) \pi ^2 -2$$ Therefore the Fourier series for f(t) is: $$\frac{\pi ^ 2}{3} + \sum \limit_{n=1} ^{\infty} A_n cos(nt) + B_n sin(nt)$$ Could someone please me if my calculations are correct? Sincerley Yours Hummingbird Last edited: May 21, 2006 2. May 21, 2006 ### benorin you may check here that the Fourier coefficients ought to be $$A_0=\frac{1}{\pi}\int_{-\pi}^{\pi}f(t) dt$$ $$A_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos (nt) dt$$ and $$B_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\sin (nt) dt$$ so that for $$f(t)=t^2$$ we have $$A_0=\frac{1}{\pi}\int_{-\pi}^{\pi}t^2 dt = \frac{2}{\pi}\int_{0}^{\pi}t^2 dt = \frac{2\pi ^3}{3}$$ where the second move is from the integral of an even function over a symmetric interval (i.e. [-a,a]) is twice the integral over [0,a], also note that the same applies to An, hence $$A_n=\frac{1}{\pi}\int_{-\pi}^{\pi}t^2 \cos (nt) dt = \frac{2}{\pi}\int_{0}^{\pi}t^2 \cos (nt) dt$$ integrate by parts twice to get EDIT: forgot to multiply by the $$\frac{1}{\pi}$$! $$A_n=\frac{2}{\pi}\int_{0}^{\pi}t^2 \cos (nt) dt = \frac{2}{\pi}\left[ \frac{2t^2}{n}\sin (nt) + \frac{4t}{n^2}\cos (nt)-\frac{4}{n^3}\sin (nt)\right]_{t=0}^{\pi} = \frac{1}{\pi}\left( \frac{2\pi ^2}{n}\sin (n\pi ) + \frac{4\pi}{n^2}\cos (n\pi )-\frac{4}{n^3}\sin (n\pi )\right)$$ $$= \frac{2\pi }{n}\sin (n\pi ) + \frac{4}{n^2}\cos (n\pi )-\frac{4}{\pi n^3}\sin (n\pi )$$ know also that $$t^2\sin (nt)$$ is an odd function, and that the integral of an odd function over a symmetric interval (i.e. [-a,a]) is zero, hence $$B_n=\frac{1}{\pi}\int_{-\pi}^{\pi}t^2\sin (nt) dt=0$$ Last edited: May 21, 2006 3. May 21, 2006 ### benorin Notably, you may perform definite integration on this http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=calculus&s2=integrate&s3=advanced [Broken] Last edited by a moderator: May 2, 2017 4. May 21, 2006 ### Hummingbird25 Hello ben, I guess it mixed up the details from a textbook example. Sorry, Then the fourier series is expressed: $$\frac{2\pi ^3}{3} + \sum \limit_{n=1} ^{\infty} \frac{2\pi ^2}{n}\sin (n\pi ) + \frac{4\pi}{n^2}\cos (n\pi )-\frac{4}{n^3}\sin (n\pi )$$ ? Sincerely Hummingbird25 5. May 21, 2006 ### benorin Rather it is expressed: $$\frac{1}{2}A_0+\sum_{n=1}^{\infty} A_n\cos (nt)+ \sum_{n=1}^{\infty} B_n\sin (nt)=\frac{\pi ^3}{3} + \sum \limit_{n=1} ^{\infty} \left( \frac{2\pi ^2}{n}\sin (n\pi ) + \frac{4\pi}{n^2}\cos (n\pi )-\frac{4}{n^3}\sin (n\pi )\right) \cos (nt)$$ 6. May 21, 2006 ### Hummingbird25 Okay I get that, By the way, f(t) has continious deriatives, and is periodic $$2 \pi$$ Then the Fourier series of f(t) converge to f(t) Uniformly on $$]-\pi, \pi[$$ ?? Or am I missing a condition? Sincerely Hummingbird 7. May 21, 2006 ### benorin I forgot to distribute the $$\frac{1}{\pi}$$ in the calculation of A_n in my first post, I fixed it: look for the EDIT, very important simplifications: $$\sin (n\pi ) = 0\mbox{ for }n=1,2,3,...$$ and $$\cos (n\pi ) = (-1)^{n}\mbox{ for }n=1,2,3,...$$ and hence $$A_n = \frac{2\pi }{n}\sin (n\pi ) + \frac{4}{n^2}\cos (n\pi )-\frac{4}{\pi n^3}\sin (n\pi ) = \frac{4}{n^2}(-1)^{n}$$ so the series becomes $$\frac{\pi ^3}{3} + \sum \limit_{n=1} ^{\infty} \left( \frac{2\pi }{n}\sin (n\pi ) + \frac{4}{n^2}\cos (n\pi )-\frac{4}{\pi n^3}\sin (n\pi )\right) \cos (nt ) = \frac{\pi ^3}{3} + \sum \limit_{n=1} ^{\infty} (-1)^{n}\frac{4}{n^2}\cos (nt )$$ that is $$\boxed{t^2 \sim \frac{\pi ^3}{3} + 4\sum \limit_{n=1} ^{\infty} (-1)^{n}\frac{\cos (nt)}{n^2} = \frac{\pi ^3}{3} -4\left( \frac{\cos (t)}{1^2}-\frac{\cos (2t)}{2^2}+\frac{\cos (3t)}{3^2}-\mdots \right)}$$​ 8. May 21, 2006 ### benorin Uniform convergence can be proved by the Weierstrass M-test, just note that $$\left| (-1)^{n}\frac{\cos (nt)}{n^2} \right| \leq \frac{1}{n^2}=M_n$$ for all n and $$\sum \limit_{n=1} ^{\infty}M_n=\sum \limit_{n=1} ^{\infty}\frac{1}{n^2}$$ converges, so the given series converges uniformly on $$\left[ -\pi ,\pi \right]$$ by the Weierstrass M-test. 9. May 21, 2006 ### Hummingbird25 Okay thank You then I only have one final question. Given the two series $$\sum_{k=1} ^{\infty} \frac{1}{k^4}$$ and $$\sum_{k=1} ^{\infty} (-1)^{k-1} \frac{1}{k^2}$$ I need to find the sum of these two. In series number 1: I can see that by the test of comparison, that it converges $$\frac{1}{k^{2+t}} < \frac{1}{k^{2}}$$ But what is the next step in finding the sum here? In series two: What do I here? Do I test for convergens ? Sincerely Hummingbird

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The microbes living on Earth are so plentiful as to be innumerable. Untold. Countless. Not in the hyperbolic sense, but the literal, gobsmacking sense. "It's estimated there are 100 million times as many bacteria as there are stars in the universe," says microbiologist Rob Knight, director of UC San Diego's Center for Microbiome Innovation. "And we know almost nothing about most of them." To map the planet's microbiome—to classify its sundry members and fathom their relationships—would be beyond ambitious. "It’s a crazy idea. Cataloguing the microbial diversity is an immense problem, you know, because there are approximately a trillion species on the planet," says microbiologist Jack Gilbert, director of the University of Chicago's Microbiome Center. It's funny to hear Knight and Gilbert talk this way. Because seven years ago, the two of them teamed up with microbiologist Janet Jansson, director of biological sciences at Pacific Northwest National Laboratory, to found the Earth Microbiome Project, a positively massive international effort devoted to—you guessed it—cataloguing the planet's microbiome. The article itself is a textbook of ecology. Martin Blaser, director of NYU's Human Microbiome Program Today, Gilbert, Knight, Jansson, and a few hundred of their colleagues unveiled the inaugural version of that microbial map: the first reference database of bacteria colonizing the planet. To do it, they developed new protocols, analytical methods, and software for identifying and comparing microorganisms collected from every continent. All told, EMP collaborators collected 27,751 samples from organisms and environments around the world, including the human gut, a bird's mouth, the soil of an Antarctic volcano, a river in Alaska, and the bottom of the Pacific Ocean. Published this week in Nature, the effort represents the work of upwards of 500 researchers from more than 160 institutions in 43 countries around the globe. It's the most macro study of the microscopic world ever published. "This article itself is a textbook of ecology," says microbiologist Martin Blaser, director of New York University's Human Microbiome Program, who was unaffiliated with the project. "Students in years to come will read it and say: Here’s where a lot of the rules originated—the rules of ecological relationships, the principles for how nature is organized." Those organizing principles are too numerous—and, in most cases, too nascent—to recount here. (The 27,751 samples collected for this meta analysis appear in some 100 other studies, half of which have already been published in peer-reviewed journals.) But Knight sums it up: "What was really remarkable about our findings was that this was true across different types of environments, whether we’re talking about microbes on animals, or on plants, or in saline or on non-saline communities," he says. "Even though the kinds of microbes in these environments are completely different, the ecological principles remain largely the same." And now microbiologists have a tool to dig up even more of those dynamic principles. But compiling the catalogue wasn't easy. As in previous studies, the researchers classified samples of bugs by sequencing the 16S rRNA gene, which carries unique mutations that act like a bacterial barcode. Once researchers have the sequences for all the bugs in a particular sample, they compare them all to each other and cluster bacteria into groups based on their similarities. Their identities become interdependent. That kind of interdependence is fine if you're assessing the diversity of bacteria from a small set of samples, from a specific region—but it makes it makes it difficult for researchers to compare bacteria between environments, or compare their observations to yours. "It really limits the ability to share information, or to accumulate information across studies," says microbiologist Jon Sanders, a postdoc in Knight's lab and coauthor on the Nature paper. It’s especially a problem if you’re dealing with billions of sequences—which is exactly what the Earth Microbiome Project had. Its researchers sequenced the 16S rRNA genes not from the microbes in one sample, or even a few hundred—but all 27,751 of them. This yielded some 2.2 billion sequences. To put that number in perspective, 10 years ago, Knight published what was, at the time, the most comprehensive analysis of the planet's microbial makeup. He and co-author Catherine Lozupone combined 16S rRNA sequences from 111 studies, for a grand total of 21,752 sequences. In Knight's words, the 2.2 billion sequences in this new paper represent a 100,000-fold expansion in our knowledge of the microbial world. So with the help of some clever algorithms, the researchers classified the 2.2 billion sequences not by clustering them, but by trimming each one down to a stretch of genetic code 90 base pairs long—a completely independent identifier. When the researchers were through scrubbing their data, they had 307,572 unique microbial sequences, almost 90 percent of which were undocumented in existing 16S rRNA databases. More on Microbes "I call it the true name of the bacteria," Sanders says. In practice, a true name means not having to wonder if the microbe you found in a lake in Colorado is the same as the one your colleague found off the coast of San Diego five years ago. And with EMP's catalogue, researchers can usually identify where a sample originated just by knowing the creatures living in it. "The key thing is, I can see the sequence now and it's meaningful, and I can see it again in 20 years and it will still be meaningful," Sanders says. "It gives us the ability to accumulate information about these bacteria across many, many studies going into the future." That makes the EMP database more than a resource—it's also a jumping off point. "It opens the field toward more complicated types of analysis, and serves as a masterful example of how microbiologists and biologists can work together to address much bigger questions" says Nikos Kyrpides, director of the Microbiome Data Science Group at the Department of Energy's Joint Genome Institute, who was not affiliated with the study. "We need to confront the fact that we’re living on a microbial planet, that the magnitude of work we need to do is enormous, and we can only do it if we work collaboratively." To that end, Jansson, Knight, and Gilbert say they're recruiting more contributors from around the world—to collect samples from a greater range of latitudes and elevations. To target fungi, in addition to bacteria. To sequence not just the 16S rRNA gene, but entire genomes. And to bridge EMP's catalogue with other databases, like the American Gut dataset. "There will be more to come," says Jansson. After all, there are 100 million times as many bacteria as there are stars in the universe. That means there are innumerable billions—maybe trillions—of microbes left to meet.

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Did you recognize that 99 is the amount of the cubes of three consecutive integers: 99 = 23 + 33 + 43? Also, 99 is an odd composite number. The comprises the factors other 보다 1 and also itself. Let’s learn about the components of 99, determinants of 99 in pairs, element factorization the 99, prime factors of 99, and also the element tree that 99. You are watching: 99 as a product of prime factors Factors that 99:1, 3, 9, 11, 33, and also 99Prime factorization of 99: 3 × 3 × 11 1 What room the determinants of 99? 2 How to calculate the factors of 99? 3 Tips and Tricks 4 Factors of 99 by prime Factorization 5 Factors of 99 in Pairs 6 Important Notes 7 FAQs on components of 99 Example: We deserve to use department to inspect whether 11 and 10 are factors that 99. 99 divides into 11 equal components with no remainder. On splitting 99 into 11 components we gain a totality number, i.e., 9. Therefore, 11 is a element of 99.99 does not get divided by 10 right into equal whole parts. On splitting 99 into 10 parts, we gain a spring number, i.e., 9.9. Therefore, 10 is no a variable of 99.Since 99 is a composite number, it will have more than 2 factors. 99 has actually a full of 12 factors. There space two typical methods to find the components of 99: Division technique and Prime factorization method .Let united state calculate factors of 99 using the division method. While considering numbers that have the right to divide 99 without remainders, we begin with 1, then inspect 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, etc. Approximately 45 (which is almost half that 99). 99 ÷ 1 = 99, Remainder = 0 99 ÷ 3 = 33, Remainder = 0 99 ÷ 9 = 11, Remainder = 0 99 ÷ 11 = 9, Remainder = 0 Tips and also Tricks 99 = 3 × 3 × 11 = 32 × 11To obtain the total numbers of components of 99, simply include one to the exponent 2 and 1 and also multiply your sums. (2 + 1) x (1 + 1) = 3 x 2 = 6 Factors the 99 = 1, 3, 9, 11, 33, and 99 Prime administer is to express a composite number as the product that its prime factors. A factor tree is a special branching diagram whereby we discover the components of a number, then the factors of those numbers, and also so on until us cannot element anymore. The ends of a aspect tree space all the prime factors of the initial number. Pair components of the number 99 space the entirety numbers which main point to get the initial number, i.e., 99. Pair factors could be either positive or an unfavorable but no a fraction or decimal number. There room 6 factor pairs of 99, and they are (1, 99), (-1, -99), (3, 33), (-3, -33), (9, 11), and (-9, -11). Important Notes 99 is a composite number.3 and also 11 are the just prime numbers that are factors of 99.The variety of factors the a offered number is finite; 99 has a complete of 6 factors. Example 1: Emma asks she students to give her the product of every the unique prime factors of the number 99. If the student give the exactly answer, what must have been your answer? Solution: Prime components of 99 space 3 and also 11. See more: Individuals Who Regularly Exercise Too Hard, Too Often Are Likely Candidates For Product the these prime components = 3 x 11 = 33 The product of all the prime factors of the number 99 is 33. Example 2: Jason is make the efforts to calculation the mean of all the components of 99. That is a bit confused; can you help him with the same? Solution: Factors the 99: 1, 3, 9, 11, 33, and 99 Sum of every the given factors = 1 + 3 + 9 + 11 + 33 + 99 = 156 To calculation the mean, we will divide the sum of the factors through the complete numbers that factors, i.e., 6. 156 ÷ 6 = 26

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Share # Two Points a and B Are on the Same Side of a Tower and in the Same Straight Line with Its Base. the Angles of Depression of These Points from the Top of the Tower Are 60° and 45° Respectively. If the Height of the Tower is 15 M, Then Find the Distance Between the Points. - CBSE Class 10 - Mathematics #### Question Two points A and B are on the same side of a tower and in the same straight line with its base. The angles of depression of these points from the top of the tower are 60° and 45° respectively. If the height of the tower is 15 m, then find the distance between the points. #### Solution Let CD be the tower. A and B are the two points on the same side of the tower. In ΔDBC tan 60^@ = (DC)/(BC) => sqrt3 = 15/(BC) => BC = 15/sqrt3 => BC = 5sqrt3 m In ΔDAC tan 45^@ = (DC)/(AC) => 1 = 15/"AC" => AC = 15 m Now AC = AB + BC :. AB = AC - BC = 15 - 5sqrt3 = 5(3 - sqrt3) m Hence, the distance between the two points A and B is 5(3 - sqrt3) m Is there an error in this question or solution? #### Video TutorialsVIEW ALL [5] Solution Two Points a and B Are on the Same Side of a Tower and in the Same Straight Line with Its Base. the Angles of Depression of These Points from the Top of the Tower Are 60° and 45° Respectively. If the Height of the Tower is 15 M, Then Find the Distance Between the Points. Concept: Heights and Distances. S

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Welcome to Let’s Blog. We’re looking forward to sharing with you the most innovative practices – methods you can use immediately in your online or your face-to-face lessons. Diversity is about all of us… …and about all of us having to figure out how to walk through this world together. Diversity: What types of diversity are there? How can we create a more inclusive classroom? What resources are out there that can really help us! Excellent questions. As usual. Let’s examine them one at a time so that in the end, you’l hopefully feel more in control of the crazines that sometimes takes over your classroom. What types of diversity are there? Well, the truth of it is that, the minute one of your students walks into the classroom, you now have diversity. You and your student probably have different learning styles, different expectations, different interests, and different speeds of assimilating information. Just being in the same space as your young learner, you are looking in the face of diversity! When the second student walks in, the diversity doubles, and so on. But…you’re used to thinking about more blatant emotional and physical differences, right? Yes, of course. Those, too. The list you’re thinking of is probably similar to what you see below. They are the categories normally delineated on the American continents: - Emotional disturbance - Hearing impairment - Intellectual disability - Multiple disabilities - Orthopedic impairment - Other health impairment (includes ADHD) - Specific learning disabilities (includes dyslexia, dyscalculia, dysgraphia, and other learning differences) - Speech or language impairment - Traumatic brain injury - Visual impairment, including blindness However, there are so many more differences that factor into classroom diversity. In fact the INEGI (Instituto Nacional de Estadística, Geografía e Informática) divides disabilities into 9 groups and 999 subgroups; some of these subgroups have lists of hundreds of disabilities in each one! So…there are a lot of elements that are included in educational diversity. Because of this, our main goal – as teachers and students – needs to be to find ways of creating learning environments that benefit everyone. In this way, we will all feel more confident about learning, and we’ll all become more understanding and kinder citizens of the world. How do we do this? Well, you probably know from the other Richmond blogs, that we’re going to go right to the sourcek – the brain – to get some grounding on what’s really going on behind diversities. Here goes – short and sweet: The factors you see in the chart below, are the major elements that affect how the brain responds to exterior stimuli. You’ll see that neurological development is affected by both biological and societal factors…and since we all have different internal and external experiences, it’s logical that we’re going to externalize different physical and emotional responses. This means more types of diversity in our classrooms. And there’s more. Diversity is affected by differences in: - frequent or sudden moves (immigration, employment change or unemployment, etc.) - language/language level - learning style - sexual orientation (and acceptance of whatever it is) - physical (height, hair color, attractiveness, cleanliness, etc.) During the recent crisis, more elements were introduced in how students acted in the learning environment: - physical and emotional isolation - violence in the home - no internet connection As teachers, it was difficult for us to assimilate all of this at once. But there are definitely techniques we can use now and in the future to balance out all of these differences, and so achieve more inclusion in our student population. Here are a few tips that address the most glaring physical problems in your physical or virtual classes. (In the next blog – Let’s Blog Diversity and the Growth Mindset, we’ll share more concrete activities.) with AUTISTIC STUDENTS: - use eye contact to initiate and maintain interactions (in virtual classes, you’ll need parental support so that the students look at the camera and so see you looking at them) - be clear and concise with instructions and explanations - give warnings about upcoming transitions and changes - give daily and weekly schedules to increase predictability - make factual information regarding fear or anxiety arousing situations obvious (Ex. ‘This is what we know about the virus. There is no reason to believe that it can get worse.’) - establish an area within the classroom where the student can get away from other students, when necessary (in virtual classes – the student may need to turn off the sound frequently to be able to shut out all the stimuli for a time) with ADD/ADHD STUDENTS: - whenever possible, let them stand and/or move around the room. They benefit from the physical movement and can usually focus better following a time of even limited exercise. (in virtual classes – encourage them to move around the room they’re in whenever they feel the need. They can turn the volume up so that they don’t miss what is happening in the class.) - use timers, taped time signals, or verbal cues to show how much time there is remaining for an activity - pair the student with a “study buddy”—a kind and mature classroom peer who can help give reminders or refocus the students when they gets off-track (in virtual classes – find a way to connect these students with the necessary students – either in a Breakout Room, by cell phones, or through other technology) with DOWN SYNDROME STUDENTS: - allow extra time for them to complete tasks - provide increased opportunities for practice - encourage activities that target muscle development (in virtual classes – send exercises to the parents for these students to do at home, such as fine motor development, use wrist and finger strengthening activities, and multisensory activities.) - Place the student at the front of the class (virtual classes – mention their name more often to keep their attention) - use gestures and expressions - use visual aids (e.g., write on the board) - rephrase and repeat questions or instructions often. What we also need to do more often (and it’s completely understandable why we don’t because we’re frustrated, tired, feel isolated, we lose patience) is to give our students compliments. We’re so used to focusing on what our diverse students do badly, or how they break the rules, break the peace in the classroom, or disrupt the class in other ways, that we forget that they do actually have moments of positive involvement. If we only remembered to give them compliments more often, we’d also see radical changes in their interactions and self-image. Here are little motivators we can introduce whenever possible: You’ll find a document attached with resources that can help you even further. In the end, however, what the experts say is the most important element we can use to achieve a more inclusive learning environment – and what you probably already use – is empathy and patience. So that’s a quick introduction to different types of diversity, and ways to make our class more inclusive – either in the physical or virtual classroom. in the next blog, we’ll share concrete activities that can also help to bring about inclusion and more community. See you there soon! In the meantime… please share the changes you see in your learning environment as you use the techniques above to create a more inclusive classroom!! Resources in English: Dweck, Carol (2017). Changing the Way You Think to Change Your Mindset Education for Persons with Disabilities Case Studies – Special needs students Fields, Donna (2018) 101 Scaffolding Techniques for Language Teaching and Learning. Supporting Students with Down Syndrome in your Classroom TASC Wheel (Belle Wallace) Recursos en español: Clasificación de tipo de discapacidad Educación de niños y jóvenes con discapacidades Educación inclusiva (2019) ‘Educación 3.0 Líder informativo en innovación educativa’ No. 33 Invierno 2019, Educaria. El prohombre (parábola sobre elecciones del tipo de inclusión) Fields, Donna (2017). Echando una mano: 101 andamiajes para el profesorado Niños superdotados: una aproximación a su realidad

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Credit: Domae M. et al., Neuroscience Letters, June 8, 2019 Many nocturnal animals including insects use a species-specific smell, that is sex a pheromone, to locate and attract potential mates. For example, female American cockroaches emit sex pheromones called “periplanones” with unique chemical structures. Males that detect them with their antennae orientate towards the pheromone source, preform courtship rituals, and mate. The Turkestan cockroach, Blatta lateralis, also known as red cockroach, has now growing attention as a world-wide invasive pest, especially in the southern United States. Its origin is temperate to subtropical zones of the Middle East. “The Turkestan cockroach is popular as live food among reptile breeders and can be easily purchased online. So, this would be the first species that expands its habitat via the internet,” says Hiroshi Nishino of Hokkaido University. Recent molecular genetic studies have shown that the Turkestan cockroach is phylogenetically close to the American cockroach in the genus Periplaneta, despite their morphological and habitational differences. This study, led by Hiroshi Nishino and published in Neuroscience Letters, found that the Turkestan cockroach uses periplanones or similar substances as sex pheromones. The experiments showed a male Turkestan cockroach has an extremely large glomerulus that specifically processes sex pheromones in a part of the brain called the antennal lobe. The glomerulus was three times bigger than that of American cockroaches, suggesting it has more sensory cells for processing sex pheromones. Accordingly, the output neuron from the large glomerulus was extremely sensitive to periplanones. As little as 0.1 femtograms of pheromone contained in a filter paper was sufficient to excite the output neuron when the paper contacted a very small region (approx. 1mm) of the antenna. This sensitivity to periplanone was more than 100 times higher compared to that of the American cockroach. Researchers also found that a male Turkestan cockroach was attracted to periplanone, but, unlike American cockroaches, the pheromone alone did not trigger courtship rituals. Courtship rituals started only after coming in contact with a female of the same species. This implies that a low volatile substance on the female’s body plays an important role in preventing the species from mating with other species. “Turkestan cockroaches adapt to inground containers where odor molecules diffuse very slowly and this could be why they have evolved an enlarged glomerulus to detect tiny packets of pheromones,” says Hiroshi Nishino. He added, “We need exercise caution when handling Turkestan cockroaches because they are a member of the genus Periplaneta, in which most have become household pests worldwide. Related Journal Article

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Geophysical monitoring of the ground above active supervolcanoes shows that it rises and falls as magma moves beneath the surface of the Earth. Silica-rich magmas like those in the Yellowstone region and along the western margin of North and South America can erupt violently and explosively, throwing vast quantities of ash into the air, followed by slower flows of glassy, viscous magma. But what do the subterranean magma chambers look like, and where does the magma originate? Those questions can’t be answered directly at modern, active volcanoes. Instead, a new National Science Foundation (NSF)-funded study by University of Wyoming researchers suggests that scientists can go back into the past to study the solidified magma chambers where erosion has removed the overlying rock, exposing granite underpinnings. The study and its findings are outlined in a paper published in the June issue of American Mineralogist, the journal of the Mineralogical Society of America. “Every geology student is taught that the present is the key to the past,” says Carol Frost, director of the NSF’s Division of Earth Sciences, on leave from UW, where she is a professor in the Department of Geology and Geophysics. “In this study, we used the record from past to understand what is happening in modern magma chambers.” One such large granite body, the 2.62 billion-year-old Wyoming batholith, extends more than 125 miles across central Wyoming. UW master’s degree student Davin Bagdonas traversed the Granite, Shirley and Laramie mountains to examine the body, finding remarkable uniformity, with similar biotite granite throughout. “It was monotonous,” says Bagdonas, who worked on the project with Frost. “Only minor variations were observed in granite near the roof and margins of the intrusion.” This homogeneity indicates that the crystallizing magma was generally well-mixed. However, more subtle isotopic variations across the batholith show that the magma formed by melting of multiple rock sources that rose through multiple conduits, and that homogenization was incomplete. Studies of the products of supervolcanoes and their possible batholithic counterparts at depth are a vibrant, controversial area of research, says Brad Singer, professor in the Department of Geoscience at the University of Wisconsin-Madison. He says the research by Frost and her colleagues offers “a novel perspective gleaned from the ancient Wyoming batholith, suggesting that it is the frozen portion of a vast magma system that could have fed supervolcanoes like those which erupted in northern Chile-southern Bolivia during the last 10 million years. “The possibility of such a connection, while intriguing, does raise questions. The high silica and potassium contents of the Wyoming granites differ from the bulk magma compositions erupted by these huge Andean supervolcanos. This might mean that the Wyoming batholith records the complete solidification of potentially explosive magma at depth, without the eruption of much high-silica rhyolite,” Singer says. “Notwithstanding, this paper will certainly provoke a deeper look into how ancient Archean granites can be used to leverage understanding of the ‘volcanic-plutonic connection’ at supervolcanoes.” Large bodies composed solely of biotite granite are more common in the Neoarchean eon (2.8 billion-2.5 billion years ago) than in younger terrains. The reason may relate to higher radioactive heat production in the past, which provided the power to drive extensive granite formation, the UW researchers say. Download the HeritageDaily mobile application on iOS and Android Planet Knowledge is a FREE to watch video on demand channel available on Freeview HD (Channel 265), Youview, Samsung connected TV’s, selected smart tv’s, tablets and smartphones using Android or iOS.

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home > tools > fasteners For accounts of pre-modern screw cutting, please go here. Modern machining was born in the industrial flowering in late 18th century Great Britain. The modern lathe, capable of cutting threads with great precision, was invented in 1797 by Henry Maudsley. Even today, for most purposes there is no need for any greater precision than that achieved by Maudslay. Creating threaded fasteners became much easier, but everyone made them to his own pattern. If you lost a nut from a machine, and the shop that made it was out of business, a new nut would have to be custom made to match the existing bolt. (Read an appreciation from a great contemporary.) Maudslay took on an apprentice, Joseph Whitworth, who proved exceptionally talented. While he was with Maudslay, Whitworth invented the method for producing a true plane surface in steel, one of the fundamental operations in precision machining. He next worked at Joseph Clements, where they were trying to build Babbage's calculating engine, the first computer, and finally set up shop for himself as a toolmaker. By 1859 he had produced a machine capable of measurements to one two-millionth of an inch. Whitworth set himself the task of devising a standard for threads. He had his own ideas about what would work best, but being a pragmatist he also collected bolts from all over England, noting which sizes experience had shown to be most useful, and the results of various thread forms. In 1841 he proposed as a standard a thread form with an included angle of 55°, and the tops and bottoms of the threads rounded with a radius equal to 0.1373 times the pitch. In 1857 experience with the first proposal led Whitworth to greatly expand the original table. Due in part to the immense prestige Whitworth gained from the display of his machines at the Crystal Palace Exhibition of 1851, Whitworth's system was in general use in Great Britain by 1860. Later a second series with finer threads (BSF) was added. (For current values, see table.) Sir Joseph Whitworth. An uniform system of screw threads. Minutes of Proceedings of the Institution of Civil Engineers, 1841, i, 157. Engineering and Architecture Journal, 1857, page 262; 1858, page 48. Americans experienced the same problems from lack of thread standardization that Britain did. The challenge was taken up by William Sellers, scion of an eminent family of American “mechanicians,” whose grandfather had made the plates with which the Continental Congress printed its currency. To William himself, among other things, we owe the color “machinery gray.” When others were decorating their machinery, he insisted on painting his a uniform light gray, in order not to obscure the functions of the parts. Sellers specified a thread form and a graded series of nuts and bolts that used it. A system of screw threads and nuts. Journal of the Franklin Institute, volume 47, page 344 (May 1864). See the same journal, volume 49, page 53 (1865), for the report of the committee, recommending the adoption of Seller's system. In 1864, a committee of the Franklin Institute recommended the adoption of Seller’s system of screw threads. The thread form became known as the “Franklin thread,” or, more commonly “Seller's thread,” and later as the “United States Standard Thread.” It became the basis of the French standard thread, and then of the Système International thread. In May 1924 it was designated the “American Standard Thread.” The main difference between Seller's thread form and Whitworth's is that the tops and bottoms of the threads (the crests and roots) are flattened. The flattened roots was a bad choice. Such angular joins in metal concentrate stress, and the process of manufacture results in high stresses at the roots of threads anyway. The result is cracks and broken fasteners. This problem was not so noticeable in Seller’s day for two reasons. One was that most machinery was stationary and the weight of a bolt rarely mattered. If a bolt broke it could be replaced with a larger one. The second reason was that thread roots tend to be rounded anyway as the tools that make the bolts become worn. With airplanes, “just put in a bigger bolt” was not a satisfactory solution, and aerospace engineers finally introduced an American thread form (UNJ) with rounded roots. For example, by changing to this thread form an American car manufacturer finally solved a persistent problem with connecting rod breakage. Manufacturers adopted Seller's thread form but rejected other parts of his system, such as the formulas for the size of square and hexagonal nuts and bolt heads, and they chose to use a different number of threads per inch for the 11⁄16 inch and 15⁄16 inch bolts. In 1907 the American Society of Mechanical Engineers (ASME) defined two series that used Seller's thread, numbering the sizes by gage numbers from 0 to 30. In the series the major diameter increased by 0.013 inch with each size. The obsolete ASME gages are described in this table. Yet another ASME “special” series used the same major diameters, but assigned different thread frequencies. Copyright © 2000 Sizes, Inc. All rights reserved. Last revised: 7 June 2007.

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(E26) AC Impedance Going Beyond Ohm's LawWith alternating current (AC), what happens to Ohm's Law? For pure resistors (e.g. those in simple bread toasters) Ohm's law is the same for both AC and DC currents, with the usual caution that here is not a fundamental law of nature, just an approximation which holds for certain materials and conditions. Fluorescent lamps don't go by it, and even the filament of a flashlight bulb increases its resistance fivefold when glowing (which helps switch it on quickly, because the initial current is 5 times larger). Nonlinear elements which protect computer power inputs don't obey Ohm's law, either: in usual operation, they act like insulators, but their insulation collapses when a large voltage spike occurs (e.g. when lightning hits a power line), protecting the device. With AC currents resistors obey Ohm's law, but in addition, new circuit elements also appear, in particular capacitors and inductors (magnetic coils, including transformers). As already noted, a capacitor will let a varying voltage pass through it. If a positive voltage rises on one plate or set of plates in the capacitor, the other plate will at first match it with an equal rise (until enough current has flowed to charge the capacitor). This makes the current effectively go through, and with AC such variations occur all the time. A capacitor's ability to transmit AC voltage depends on the speed of variation: radio-wave frequency are transmitted quite easily (the wave duration is too short for the circuit to supply much charge), sometimes evev unintentionally when circuits operate close to each other (your computer probably shields some of them inside grounded aluminum boxes, to block such "pick-up"). House-current AC varies relatively slowly, 60 cycles per second, and uses larger capacitors--but smaller ones serve aboard ships and airplanes, with 400-cycle AC). Constant voltages of steady DC are, of course, completely blocked. RadiansA "simple" AC voltage rises and falls in a wave like a the trigonometric sine function (red) or cosine function (blue, differing only by a horizontal shift). A location on the wave at any instant is called the AC phase (x in the graph) and is often measured in degrees, 360° for a full cycle, a system inherited from ancient Babylonians, who noted that the position of the Sun in the sky, (relative to the background stars), makes each year a full circle and advances each day about one degree. Then if f is the AC frequency in cycles per second, during a time t the functions go each second through f cycles. In mathematics, however, angles are frequently measured in radians, units equal to about 57°. One full rotation contains 2 π radians, where π=3.1415926... is the ratio between the circumference of a circle and its diameter (yes, I have a super motorbike to travel about the roads foolishly...). It may seem strange to measure angles in units of which the circle does not even contain a whole number and which can only be used to some approximation. However, radians give mathematical advantages, e.g. formulas such as where 3!=6 ("three factorial") is produced by multiplying all integers up to 3, also 5!=120 by multiplying up to 5, and so on. That formula holds in radians; in texts using degrees, each x on the right must be multiplied by 2π/360. Following most texts, we work here in radians (except that phase angles are sometimes in degrees), and in place of the frequency f cycles-per-second, use the "angular frequency" ω = 2 π f radians-per-second, denoted by the lower-case Greek letter omega (upper case Ω is used for "ohms"). AC Impedance--CapacitorsA capacitor of C farad with charge Q coulomb stores electric energy The necessity to "load up" the charge in the beginning of the AC cycle (when the sine function starts fom zero) and then "unload" it again impedes the flow of current. Circuit analysis therefore includes an impedance A capacitor with large plates (and large C) has a smaller impedance, since it lets AC pass more easily--if a positive charge is added on one plate, a corresponding charge quickly accumulates on the other plate, even at moderate voltage changes, and the voltage signal continues on the other side. And the impedance of a capacitor decreases as the angular frequency ω grows, meaning a higher frequency can more easily propagate through space than a lower one. Steady DC with ω=0 cannot propagate at all but is stopped by the capacitor, which for DC is just a break in the continuity of the circuit. AC Impedance and Admittance--InductorsAn opposite behavior is observed in current carrying coils, where every AC cycle need invest energy to build up a magnetic field, only getting the energy back when the current drops and reverses. This energy-storing property is called inductance (or "self-inductance") and is especially noted in coils, magnets and transformers: just as capacitance stores electric energy, inductance stores magnetic energy. Its unit is names the henry after Joseph Henry who invented the practical electromagnet, and inductances in such units are usually denoted by the capital letter L. In analogy to an earlier equation, an inductor of L henry carrying a current of I amperes stores magnetic energy If you suddenly break a circuit which includes a large inductance, this may create a big spark from the high voltage generated by the magnetic energy, as the current tries to keep going. The impedance of a device with stored magnetic energy (coil, electromagnet, transformer, electric motor) has the opposite frequency dependence than capacitance. It is low at low frequency--for DC, only the ohmic resistance of a coil resists the current--but gets larger the more rapid the change, e.g. the higher the frequency. Its value is The graphic symbol for an inductance is a spiral, denoting a coil--the spiral has a few parallel straight lines next to it if the coil has an iron core, and two parallel coils with a bundle of straight lines in the middle to denote an iron-core electric transformer. Just as a resistance R is mirrored by conductance 1/R, so impedances have corresponding AC admittances Cω and 1/Lω . Where complications beginIt would have been nice if we could carry over all the formulas of resistor circuits to capacitors and inductors. In some simple cases, that is possible: two capacitors (C1, C2) in parallel act like a single capacitor (C1 + C2), and two inductors (L1, L2) in series are equivalent to a single one (L1 + L2). The complimentary equations also hold, but please note that capacitances in (series, parallel) add up like resistors in (parallel, series). Things grow complicated in circuits including both types, and also resistors. Impedance not only depends on angular frequency ω, it also can shift the phase of the AC wave forward or backward in time (or in angle, if you wish). The sine waves of volts and amperes may shift their starting points in the cycle and that, at the very least, reduces the power transmitted. Such shifts can be represented by replacing sin ωt with combinations of sines and cosines, or by a mathematical scheme which requires "complex numbers", a set of mathematical objects broader than "ordinary" numbers, including all of those plus "imaginary" parts proportional to multiples of i=√–1 (in texts on electricity, it is often replaced by the letter j, reserving i for current). For this introductory overview, that would be going too far; you may look up here, a file which is part of an extensive course on electricity. Let us just note that a mere shift of phase (when ohmic resistance is negligible) absorbs no energy, it just mixes up some of the waves because of temporary storage of electric energy (in capacitors) or magnetic energy (in inductors). Such combinations can serve as frequency filters, useful (for instance) when you try to tune in a radio or TV broadcast of a certain frequency while rejecting all others which are "on the air" at the same time. The Crystal Radio When tuning a radio, the knob which you turn is usually connected to a variable capacitor, whose symbol is a pair of parallel lines (like any capacitor) crossed by a slanted arrow. Like any capacitor, this one too has two sets of plates, separated by empty space or by an insulating "dielectric material." One set is fixed (and often grounded), while the other one is connected to the shaft of the tuning knob. By turning the knob, you change the extent to which the plates overlap--and since only the overlap (essentially) contributes capacitance--this changes the effective capacitance. Consider a capacitor and inductor connected "in parallel" between points A and B. If the inductor has negligible ohmic resistance, for a steady DC (ω=0) it essentially short-circuits A to B, so their voltage difference is zero. On the other hand, for DC the capacitor is an open gap and can be omitted from the circuit. Going to the other extreme, a high radio frequency (say, ten million cycles per second), the capacitor places a very low impedance between A and B--it acts like a short circuit for the radio frequency. The coil on the other hand has such high impedance that we may just as well ignore it. At these extremes of the frequency range, the impedance between A and B is near zero--like a direct connection which allows no voltage drop. But not in a range of ω in-between, where both capacitance and inductance have non-zero impedance. In fact, somewhere in the middle range a resonance exists and the impedance is large, so that if B is earthed while A receives a large AC wave, an AC voltage can build up between these points. Jump ahead a few steps. Suppose A is attached to a wire in space, suspended from insulating supports and subject to whatever electric fields exist there. If the location is near a strong AM radio station, it can capture some of the radio signal. The wire then becomes a receiving antenna, marked in the diagram by several short lines forming moderate angles (like a schematic broomstick). If the variable capacitor is now adjusted so that the resonant frequency f of the circuit matches the frequency of the transmitter, the voltage of A will rise and fall with the radio signal. If the resonance is elsewhere, the impedance between A and B is small and only a small signal voltage exists between A and B. However, we not only want to detect the radio-frequency signal, but also decode the sound signal it carries. One of the earliest methods of encoding sound was amplitude modulation (AM), taking a wave with fixed frequency and modulating its height to follow the electric signal from a microphone. Car radios still receive AM radio, and so will the outdoor antenna we described. If we now insert a solid state diode like the one converting AC to DC in section #16, the earphones may respond to the average voltage, filter away the radio frequency and leave speech or music only. You will then have created a "crystal radio" like the one built by schoolboys in the early days of radio. Reliable "solid-state rectifiers" did not yet exist, but certain crystals touched by thin wire "whiskers" had similar properties, and persistent amateurs could sometimes hear broadcasts. It wasn't a good radio receiver then, and even now, with a solid state rectifier, it performs poorly, because the earphones have no energy source except the captured part of the radio wave, which tends to be quite weak. Regular radios amplify the weak signal, using the energy of batteries or of power supplies like the one in section #16. But the crystal radio was one beginning. Radio Waves themselves will be discussed later. For now, just take for granted that all space around you contains oscillating voltages, broadcast by countless antennas in many frequencies, all over the country and the world. Next Stop: E27. Electro-Magnetic Waves, at last!| Author and Curator: Dr. David P. Stern

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WHAT IS PGS? Pre-implantation genetic screening (PGS) is a laboratory procedure performed as part of an IVF cycle where a genetic assessment (or screening) of the embryos is carried out. This information is used to exclude from transfer to the uterus, embryos carrying certain genetic errors. In some cases PGS involves examining the chromosomes of the embryo for numerical or structural errors, in other cases PGS involves testing for a known single abnormal gene. WHAT ARE CHROMOSOMES AND GENES? Chromosomes are structures found in the centre, or nucleus, of all cells. They are made up of ‘genes’ that code all the instructions and information needed for the development and function of all the cells of the body. A human typically has an identical set of 46 (23 pairs) chromosomes in every cell. One set of 23 is inherited from the father’s sperm, the other from the mother’s egg. The two sets come together at fertilization and a copy of the full set is inherited by all of the cells of the developing embryo and subsequently by all the cells of the body. Although all the cells of the body contain a copy of the same set of chromosomes, different instructions (genes) are read from the chromosomes depending on what functions are required of that cell (e.g. liver, bone etc). Together the genes control the development and the characteristics of each individual cell; the developing embryo; and subsequently the individual. WHY DO WE CHECK THE CHROMOSOMES IN THE EMBRYOS OF SOME PATIENTS? If there are errors in the number or structure of the chromosomes in the cells of an embryo, the chances of progressing to a normal pregnancy may be significantly reduced. One condition that PGS may be used to detect is called aneuploidy. This is where there are missing or extra chromosomes present in the cells of the embryo. Aneuploidy tends to occur with increasing incidence in the eggs (and subsequently the embryos) of women as their age increases. This is thought to be one of the reasons that women’s fertility declines with age. While there is conflicting evidence as to the benefit of PGS to generally screen for aneuploidy, some couples may elect to use PGS to help identify which of their embryos has a normal number of chromosomes and hence has the best chance of pregnancy. Some patients seeking fertility treatment may themselves have particular errors in their chromosomes called translocations. This is where a section of one chromosome has broken off and attached to another, or has perhaps even been swapped with a section somewhere else. The translocation may not affect the patient themselves at all, but it may mean that not all of their sperm or eggs have an intact set of 23 chromosomes. This may affect their fertility as the majority of their embryos may not then have the normal intact set of 46 chromosomes. The abnormal embryos may not develop beyond the early stages; may not implant; or may miscarry early in the pregnancy. By using PGS to test a couple’s embryos for the presence of these translocations, we can identify and avoid transferring embryos that carry the translocation and which we know WHY DO WE CHECK SPECIFIC GENES IN THE EMBRYOS OF SOME PATIENTS? The genes that provide the instructions controlling the function and characteristics of cells can come in various forms (or alleles). Alternate forms of many genes give rise to the normal variation that we see among individuals. The gene that codes for the pigment in the human eye, for example, codes either for pigment (brown/green) or no pigment (blue) etc and as a result humans have different eye colours. Sometimes however, one form of a gene actually results in a malfunction of a particular cellular activity and a disease condition results. Cystic fibrosis and Huntington’s disease for example are diseases that are ‘single gene disorders’. Couples who know that they are at risk of passing on a single gene disorder to their offspring may elect to have IVF and to use PGS to screen their embryos for the presence of the ‘mutant’ allele and to thereby choose not to transfer the affected embryos. SO WHAT HAPPENS IN THE LAB DURING A PGS CYCLE? For a couple undergoing an IVF cycle including PGS, the process is very similar to a normal blastocyst cycle. However, what happens in the laboratory from the time between fertilization and embryo transfer on day 5 is more complex. To test the genetic status of each embryo, one or two cells are removed or ‘biopsied’ from each embryo on day 3 after insemination when most embryos are somewhere between 5 and 8 cells. This is performed at Life Fertility Clinic by an experienced embryologist using a high powered microscope and specialist micromanipulation tools. The cells are then sent to a specialist pathology laboratory for examination of the chromosomes or testing for a particular gene as appropriate. The test results generally take 24-48 hours to be returned from the pathology lab. In the meantime, the biopsied embryos are maintained in culture in the IVF laboratory and continue developing to the blastocyst stage. Once the results of the chromosome or gene analysis are received, the doctor and embryologist will use these and the embryology observations to decide which embryos are suitable for transfer or freezing. WHAT SHOULD WE BE AWARE OF BEFORE STARTING PGS? - The Biopsy procedure: It is possible that some or all of a couple’s embryos may fail to reach a stage of development where they are considered suitable for embryo biopsy. Those embryos which are not suitable for biopsy will not be tested. While every possible measure is taken to optimise the biopsy procedure, it is possible that embryos may be damaged by the procedure or fail to survive after it. Furthermore, as is the case for all blastocyst cycles, irrespective of the genetic testing results, some (or all) embryos may fail to develop to a stage that is suitable for either embryo transfer or freezing. - The Genetic Test Results: Even if the embryos are successfully biopsied, it is possible that technical or embryo related limitations may prevent results being obtained for some or all of the embryos tested. In addition, PGS results can sometimes be inconclusive for some or all of the embryos tested. In either of these circumstances, couples need to consider what to do with such embryos with regard to embryo transfer, cryopreservation or allowing them to succumb. PGS results are not 100% accurate and therefore despite favourable screening results, it is still possible that an affected embryo may be transferred following PGS, potentially resulting in a pregnancy and birth of an affected child. When PGS is used for single gene disorders, it will only screen for the single gene disorder in question and does not guarantee that any child resulting from a screened embryo will be free from any other abnormalities. When PGS is used for aneuploidy or translocation screening, only a subset of chromosomes is tested (13, 16, 18, 21, 22, X, Y and any specific translocation previously identified as relevant). A result of ‘no abnormality detected’ does not therefore exclude the possibility of abnormalities related to chromosomes which have not been tested which may result in a failed implantation, miscarriage, pregnancy and birth of an affected child. It is strongly recommended that patients consider undertaking either amniocentesis or chorionic villi sampling (CVS) to confirm the early embryo diagnosis if a pregnancy is established following treatment involving PGS. HOW DO WE GET STARTED ON A PGS CYCLE? PGS is carried out as part of an IVF/ICSI cycle so the first step is to speak to a fertility specialist who will organise all aspects of the IVF and PGS treatment. As part of the preparation for treatment, patients are likely to need a number of blood tests and all patients must see a Genetic counsellor. Depending on the type of genetic testing, required the preparation time for PGS can range from one to nine months. This time is required by the pathology lab to tailor the testing for each couple and to ensure that the testing procedure is as accurate as possible. For more information or queries regarding any of the services offered at Life Fertility Clinic, please contact us.

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Lymphoma is a cancer of the lymphatic system.Your lymphatic system is made up of a type of white blood cell.This is called a lymphocyte. It is located in your lymph nodes, spleen, tonsils and bone marrow (where blood cells are manufactured). The lymphatic system helps your body fight infection and disease. There are two main types of lymphoma: There are many types of treatment for lymphoma. Your treatment choices depend on three main things: You should ask the following questions when you are making a decision about your treatment: Together, you and your health care team will make a choice about which treatment (or treatments) is best for you.You should talk with your doctor, nurse, and other members of your health care team.You may ask a lot of questions about your treatment choices before making a choice. This treatment increases your body's natural ability to fight cancer. It does this by giving a boost to your immune system.There are several kinds of biologic therapy: This treatment uses drugs to kill cancer cells and reduce the size of cancer tumors. Chemotherapy drugs may also affect healthy cells and cause side effects like hair loss or mouth sores.There are many types of chemotherapy drugs. Many drugs are often used together for chemotherapy.Radiation Therapy This treatment uses radiation (high energy x-rays) to kill cancer cells.The treatment often only takes place in the part of your body where the lymphoma is located.Transplants Sometimes high doses of chemotherapy destroy the lymphoma cells and your bone marrow, which is the "factory" for blood cells. To help your bone marrow make new healthy blood cells, some stem cells (immature cells that will grow up into red blood cells, white blood cells, and platelets) may be taken with a special machine before chemotherapy is given. These cells are then transplanted (put back) into the body. These transplanted cells will then find their way to the bone marrow and restore it, so that it can build healthy new blood cells. There are two types of transplants: This means that you do not have to get any active treatment now. But, you may need to get treatments later, if tests show that your cancer is growing. Watchful waiting is usually recommended only for people with slow-growing lymphomas.Clinical Trials These are research studies that help doctors learn more about lymphoma treatment.They can also help people with cancer, because it allows them to receive the treatment. Often, clinical trials are the only way patients can receive new treatments, which are not otherwise available. Clinical trials can help doctors learn about: While clinical trials can provide many benefits, they can also be harmful for some patients.You should speak with your doctor, nurse, or health care team about clinical trials.Top of Page

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## Intermediate Algebra (6th Edition) $8xy^{2}+2x^{3}+3x^{2}-3$ We are given the expression $(-3+4x^{2}+7xy^{2})+(2x^{3}-x^{2}+xy^{2})$. We can combine like terms to simplify. $(7xy^{2}+xy^{2})+2x^{3}+(4x^{2}-x^{2})-3=8xy^{2}+2x^{3}+3x^{2}-3$

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### A2 Ch 6 Polynomials Notesx ```Algebra 2  Warm Up     A monomial is an expression that is either a real number, a variable or a product of real numbers and variables. A polynomial is a monomial or the sum of monomials. The exponent of the variable in a term determines the degree of that term. Standard form of a polynomial has the variable in descending order by degree.  The degree of a polynomial is the greatest degree of any term in the polynomial  Write each polynomial in standard form and classify it by degree. You can write a polynomial as a product of its linear factors  You can sometimes use the GCF to help factor a polynomial. The GCF will contain variables common to all terms, as well as numbers  If a linear factor of a polynomial is repeated, the zero is repeated. A repeated zero is called a multiple zero. A multiple zero has multiplicity equal to the number of times the zero occurs.   page 323 (1-11, 17-35)odd you do NOT need to graph the functions.    Two people per worksheet. Take turns at each step, first partner decides what you multiply the divisor by, second partner agrees and does the multiplication, first partner agrees and does the subtraction, then switch for next term. You may do the work on the worksheet, paper or the white board. If you use the white board you must have me check EACH answer as you complete it. Warm Up: 1. Write a polynomial function in standard form with zeros at -1, 2 and 5. 2. Use long division to divide:  3. Use long division to divide  Solve for all three roots    Homework: page 330 (227-33) odd page 336 (13 – 31) odd,  Solve these equations:  1. x3 + 125 = 0  2. x4 + 3x2 – 28 = 0  To find all the roots of a polynomial: ◦ determine the possible rational roots using the rational root theorem (ao/an) ◦ Use synthetic division to test the possible rational roots until one divides evenly ◦ Write the factored form and solve for all roots  Use the quadratic formula if necessary  You may need to use synthetic division more than once     Warm Up Find the polynomial equation in standard form that has roots at -5, -4 and 3 Find f(-2) for f(x) = x4 – 2x3 +4x2 + x + 1 using synthetic division Solve x4 – 100 = 0   Practice Problem: List all the possible rational roots of ◦ 3x3 + x2 – 15x – 5 = 0 Use synthetic division to determine which of these is a root Factor and solve for the rest of the roots of the equation.  A third degree polynomial has roots 2 and √3. Write the polynomial in standard form.   Homework p 345 (11-23) odd  A selection of items in which order does not matter is called a combination   homework p 354 (1-29) odd       Warm Up Find the zeros of the function by finding the possible rational roots and using synthetic division. multiply each and write in standard form: (x + y)2 (x + y)3 (x + y)4     Notice that each set of coefficients matches a row of Pascal’s Triangle Each row of Pascal’s Triangle contains coefficients for the expansion of (a+b)n For example, when n = 6 you can find the coefficients for the expansion of (a+b)6 in the 7th row of the triangle. Use Pascal’s Triangle to expand (a+b)6  If the terms of the polynomial have coefficients other than 1, you can still base the expansion on the triangle. Warm up  To find a particular term of a binomial expansion you do not need to calculate the entire polynomial! Ex: Find the 5th term of (x – 4)8  Find the 4th term of (x – 3)8    Chapter 6 Test this Thursday (5th) or Friday (4th) Homework: Complete practice test ```

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1 October, 03:25 # How many positive integers less than 2018 are divisible by at least 3, 11, or 61? +2 1. 1 October, 04:56 0 To find all the positive integers less than 2018 that are divisible by 3, 11, and 61, you will use what you know about factors. 3, 11, and 61 are all answers. So are 33, 183, 671, and 2013. If you put these in factors, the product will be divisible by them! 3 x 11 = 33 3 x 61 = 183 11 x 61 = 671 3 x 11 x 61 = 2013 Take each number and square it, cube it, etc ... 9, 27, 81, 243, 729 121, 1331 9 x 11 = 99 27 x 11 = 297 81 x 11 = 891 121 x 9 = 1089 121 x 3 = 363 61 x 9 = 549 61 x 27 = 1647 Everything in bold is a correct answer.

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Forests and water - case studies Watershed management and population dynamics in the Middle Hills and Terai, Nepal It is difficult for management activities to achieve their goals without a proper understanding of the many interrelated physical, biophysical and human factors that act on watersheds. Such understanding is often lacking in Nepal, where there is little evidence-based information for watershed planning. Benchmarks and changes resulting from watershed management interventions are seldom quantified, and the resource endowment and fragility of watersheds are rarely evaluated properly. Time series data for human-induced factors are lacking, and most studies do not separate natural from human causes. This gives rise to several misconceptions, among which the most important for national development is that migration and resettlement of the Middle Hills population to the Terai lowland decreased upstream degradation and improved watershed management at the river basin level. This policy was started in the late 1960s, through measures to promote the migration of Middle Hills farmers to rehabilitated lowland areas. Landless people were the primary beneficiaries of resettlement. Several projects supported the development of infrastructure and off-farm activities for income generation, and the introduction of high-yielding crop varieties and hybrid domestic animals. Most of these programmes were donor-funded and assisted by Western experts. The impact of this policy on upstream/downstream linkages in this mountainous country is arguable, however. The mass movement of people to the Terai has reduced population densities in some Middle Hills areas, and prevented the local population from growing beyond the carrying capacity, but the population of the Terai lowlands increased from 3 million in 1961 to 11 million in 2001. Resettlement of people from the Middle Hills has resulted in half the national population settling in a fragile, flood-prone, unhealthy, tropical rainforest ecosystem. At the same time, decreasing population pressure has not improved soil conversation and water management in the Middle Hills. It is estimated that between 1991 and 2001, migration caused wage labour costs to double in the Middle Hills, while the selling price of rice increased by only 50 percent. There are therefore few incentives for local farmers to maintain their paddy terraces, which are vital for both food security and watershed management. Devastating landslides and mass wasting in the Middle Hills continue to be blamed on local people overexploiting natural resources, rather than on a combination of natural events and ill-conceived policies. The floods and heavy sedimentation that affect the Terai are attributed to mass wasting in the hills and mountains, with little consideration of other human factors, such as the accumulation of sediments in downstream dam basins and irrigation channels, and intense human interference in riverbank areas. Policies are needed that appraise and manage watersheds in the light of such multi-layered and multisectoral interactions. Adapted from K. Poudel. 2005. “Watershed management in

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Science is not something mysterious. Being scientific involves being curious, observing, asking how thing happen, and learning how to find the answers. Curiosity is natural to children, but they need help understanding how to make sense of what they see. This book provides examples of the many simple activities children can do. It might even inspire them to make up their own experiments to see why and how things turn out the way they do. We can use this book to have fun with our children while they learn, and see how they enjoy the wonderful world of science.

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Nicaragua Canal - 1895-1916 The Nicaragua Canal Board, also known as the Ludlow Commission after its chairman, Col. William Ludlow, was appointed by President Grover Cleveland, April 25, 1895, to ascertain the feasibility of completing the Nicaragua Ship Canal begun by the Maritime Canal Company. The Board report, October 31, 1895, recommended a thorough reexamination of the route. President William McKinley appointed the Nicaragua Canal Commission (first Walker Commission), under Rear Adm. John G. Walker, July 29, 1897, which surveyed the route of the canal. The Nicaragua Canal [first Walker] Commission reexamineed the logistics of a canal route through the Isthmus of Nicaragua. Any canal had a powerful domestic enemy in the transcontinental railway combination. These very Pacific roads were, however, themselves the first fruits of the country's post-bellum policy of encouraging private enterprise with promises of governmental support, and now it was proposed to carry out somewhat the same idea with regard to the Nicaragua canal. Among many other projects, J.A. Latcha had planned the construction of the Duluth, South Shore, and Atlantic Railway Company (DSS&A) through the Upper Peninsula of Michigan in 1886. At that time, Latcha was the chief engineer and superintendent of construction for Brown, Howard & Co., contractors for the building of the DSS&A. Latcha argued against the Canal in 1898, writing in the a After studies from December 1897 through February 1899, the Nicaragua Canal [first Walker] Commission submitted its report in March 1899. The commission estimated the cost of construction at $118,113,790 not including interest and administration. The Senate Committee on Interoceanic Canals was established on 15 December 1899, succeeding the Senate Select Committee on the Construction of the Nicaragua Canal, 1895-99. As its name implies, the initial focus of this committee was on legislation to authorize the construction of an isthmian canal to connect the Atlantic and Pacific Oceans, not neccessarily through Nicaragua. Taking advantage of divisions within the Conservative ranks, Jose Santos Zelaya led a Liberal revolt that brought him to power in 1893. Zelaya ended a longstanding dispute with Britain over the Atlantic Coast in 1894, and reincorporated that region into Nicaragua. The Hay-Pauncefote Treaty of 1901 abrogated the earlier Clayton-Bulwer Treaty between the US and UK, and licensed the United States to build and manage its own canal. Following heated debate over the location of the proposed canal, on June 19, 1902, the U.S. Senate voted in favor of building the canal through Panama. Investors in the Panama Canal had convinced the United States Congress of the danger posed to passengers going such a short distance from an active volcano -- at that this time both the Concepcion and Momotombo Volcanoes had experienced eruptions. Unable to persuade the United States to build in Nicaragua, starting in 1904 Nicaragua's President Zelaya sought support from France, Germany and Japan to build a canal in Nicaraguae to compete with the the US-built canal through Panama. By 1909, differences had developed over the trans-isthmian canal and concessions to Americans in Nicaragua; there also was concern about what was perceived as Nicaragua's destabilizing influence in the region. In 1909 the United States provided political support to Conservative-led forces rebelling against President Zelaya and intervened militarily to protect American lives and property. With the exception of a 9-month period in 1925-26, the United States maintained troops in Nicaragua from 1912 until 1933. Nicaragua and the United States signed but never ratified the Castill-Knox Treaty in 1914, giving the United States the right to intervene in Nicaragua to protect United States interest. Emiliano Chamorro signed the treaty with Williams Jenning Bryan on 5 August 1914 in which the United States agreed to give an exclusive concession for building an interoceanic canal across the San Juan River for a period of 99 years. By signing this treaty Nicaragua received the sum of three million dollars. Because the United States had already built the Panama Canal, however, the terms of the Chamorro-Bryan Treaty served the primary purpose of securing United States interests against potential foreign countries -- mainly Germany or Japan -- building another canal in Central America. The treaty also transformed Nicaragua into a near United States protectorate. The modified version of Castill-Knox Treaty, the Chamorro-Bryan Treaty, omitting the intervention clause, was finally ratified by the United States Senate in 1916. The convention with the United States terminating the Convention respecting a Nicaragua canal route (Chamarro-Bryan Treaty) of 5 August 1914 was signed in Managua on 14 July 1970. It was approved by resolution 3 of 27 July 1970 and resolution 282 of 10 August 1970, and ratified by decree 2 of 17 February 1971. |Join the GlobalSecurity.org mailing list|

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# Product of roots of quadratic equation product of roots of quadratic equation ## Product of Roots of a Quadratic Equation To understand the product of the roots of a quadratic equation, let’s start by defining what a quadratic equation is. A quadratic equation is generally expressed in the standard form: ax^2 + bx + c = 0, where (a), (b), and (c) are coefficients, and a \neq 0 ### Solution By Steps: The roots of a quadratic equation are found using the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Here, the roots are given by: x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} 2. Product of the Roots: Let’s denote the roots by x_1 and x_2. The product of the roots can be expressed as: x_1 \times x_2 3. Derivation Using Vieta’s Formulas: Vieta’s formulas relate the coefficients of the polynomial to sums and products of its roots. For the quadratic equation ax^2 + bx + c = 0: • The sum of the roots (x_1 + x_2) is given by: x_1 + x_2 = -\frac{b}{a} • The product of the roots (x_1 \times x_2) is given by: x_1 \times x_2 = \frac{c}{a} Proof: • Start from the factored form of the quadratic equation derived from its roots. Suppose the roots are x_1 and x_2. Then, we can write the quadratic equation as: a(x - x_1)(x - x_2) = 0 • Expanding the above expression: a(x^2 - (x_1 + x_2)x + x_1 x_2) = 0 • Compare this with the standard form ax^2 + bx + c = 0: • The coefficient of (x) is -(x_1 + x_2), which implies b = -a(x_1 + x_2). This confirms the sum of the roots x_1 + x_2 = -\frac{b}{a} • The constant term is a \cdot (x_1 \times x_2), which implies c = a(x_1 \times x_2). This confirms the product of the roots x_1 \times x_2 = \frac{c}{a}. Hence, the product of the roots of the quadratic equation (ax^2 + bx + c = 0) is: x_1 \times x_2 = \frac{c}{a}

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of 20 /20 Chapter 6 Hypothesis Testing and Confidence Intervals others • Category ## Documents • view 3 0 Embed Size (px) ### Transcript of Chapter 6...Chapter 6 Hypothesis Testing and Confidence Intervals Learning Objectives • Test a... Chapter 6Hypothesis Testing and Confidence Intervals Learning Objectives • Test a hypothesis about a regression coefficient • Form a confidence interval around a regression coefficient • Show how the central limit theorem allows econometricians to ignore assumption CR4 in large samples • Present results from a regression model • We propose a value of β1 and test whether that value is plausible based on the data we have • Call the hypothesized value • Formal statement: Null hypothesis: H0: β1= Alternative hypothesis: H1: β1≠ • Sometimes the alternative is one sided, e.g., H1: β1< • Use one sided alternative if only one side is plausible *1β *1β*1β *1β Properties of b β . .[ ]s e b The z-statistic* 1 1 1. .[ ]β− =bzs e b For any hypothesis test: (i) Take the difference between our estimate and the value it would assume under the null hypothesis, then (ii)Standardize it by dividing by the standard error of the parameter If z is a large positive or negative number, then we reject the null hypothesis. We conclude that the estimate is too far from the hypothesized value to have come from the same distribution. If z is close to zero, then we cannot reject the null hypothesis. We conclude that it is a plausible value of the parameter. How large should the t statistic be before we reject the null hypothesis? ( )1 1 1 ~ 0,1. .[ ] β−= bz Ns e b If CR1-CR3 hold (plus CR4, if the sample is small) *1 1 1. .[b ]bt est s eβ− = Compare this to the t-statistic formula: • If the null hypothesis is true, the numerators of the two expressions are the same- If we knew the standard error, we would know the distribution of our t-statistic. • Knowing the distribution means that we know which values are likely and which are unlikely. • In particular, from the normal table, we would know that getting a t-statistic larger than 1.96 only happens with probability 2.5%. • Getting a t-statistic less than -1.96 also only happens with probability 2.5%. Example: California schools data 0 1 1 1: 0 vs : 0β β= ≠H H Hypothesis: free-lunch eligibility (FLE) is uncorrelated with academic performance (API) 951.87 2.11= − +i i iAPI FLE e Put this into practice with our California schools data 0 1: 0H β = (free-lunch eligibility, FLE, is uncorrelated with academic performance, API) 1 1: 0β ≠H The ideal t-statistic would use the standard error from the population, i.e., Estimate from sample of 20 schools Sd from whole population (usually don’t know this) The absolute value of this test statistic 5.15 > 1.96, so we reject the null hypothesis at 5% significance. 2.11 0 5.150.41 − −= = −z But We Don’t Usually Know the Population Variance 86.536.0 011.2−= −−=t Std error estimated from sample of 20 schools • Exceeds the critical value (|-5.86| > 2.10), so we still reject the null hypothesis. • But remember that we had to assume CR4 in order to perform this hypothesis test and construct the confidence interval. • If there’s any reason to think that the population errors are not normally distributed (picture a nice bell curve), this analysis will be very hard to defend to your colleagues. How Big a Sample is Big Enough? 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 1 2 3 4 5 6 7 or more Figure 6.1. Number of People per Household; US Census 2010Source: https://www.census.gov/hhes/families/data/households.html • This distribution is certainly not normal Distribution of the t-statistic Looks More Normal the Bigger the Sample 2.52/ bts N− = Average household size, 2010 US Census Figure 6.2. Central Limit Theorem Implies t statistic gets more Normal as N Increases CA Schools Errors Look Close to Normal -250 -200 -150 -100 -50 0 50 100 150 200 250 Figure 6.3. Histogram of Errors from Population Regression Using All 5765 CA Schools • Range of values that we can reasonably expect the true population parameter to take on. • It is the set of null hypotheses that you cannot reject. • In a large sample, this means that a 95% confidence interval is all the values for which • More common way to express the confidence interval: The 1-α% confidence interval is all the values in the range , where where c is the critical value for a two-sided test. Confidence Interval *1 1 1 1.96 1.96. .[ ]β− − ≤ ≤b est s e b *1β ( )1 1 1* . .[ ]β = −b c est s e b ( )1 1 1* . .[ ]β = +b c est s e b 1 1,β β Hypothesis Testing in Multiple Regression* . .[ ]k k k btest s e b β−= 1 2 1 = = ∑= N ki ii k N kii v yb v 2 21 . .[ ] σ = =∑k N kii s e bv •22 2 12 1 . .( ) ,1 1 = = = = =− − − − ∑∑ Nii k Nkii es SSEest s e b sN K N Kv • Compare the R2 from two regressions, one with all the RHS variables in it, the other without any RHS variables (only an intercept, which will equal the mean of Y) • Form the Wald statistic: • Kalt in the denominator is the number of X variables in the main model • The more the RHS variables “explain” the variation in Y, the bigger this test statistic will be • Under the null hypothesis, the Wald statistic has a Chi-square distribution with qdegrees of freedom, where q is the number of RHS variables omitted under the null hypothesis Do We Have a Model? 2 2 2(1 ) / ( 1)alt null alt alt R RWR N K −= − − − Example: Do We Have a Model to Predict API in CA Schools? 0.86 0 104.43(1 0.86) / (20 2 1) −= = − − −W Because 104.43 > 5.99, we reject the null hypothesis at 5% significance Degrees of Freedom Significance level 0.10 0.05 0.01 1 2.71 3.84 6.642 4.61 5.99 9.213 6.25 7.82 11.354 7.78 9.49 13.285 9.24 11.07 15.09 How to Present Regression Results 1 1 2 2 777.17 0.51 2.34(37.92) (0.40) (0.47) Sample size 200.84 = − + + = = i i i iY X X e R How to Present Regression Results 2 Variable Estimated Coefficient Standard Error t-Statistic Free-lunch eligibility –0.51 0.40 –1.28 Parents’ education 2.34 0.47 5.01 Constant 777.17 37.92 20.50Sample size 20R2 0.84 What We Learned• We reject the null hypothesis of zero relationship between free lunch eligibility (FLE) and academic performance. • Our result is the same whether we drop CR4 and invoke the central limit theorem (valid in large samples) or whether we impose CR4 (necessary in small samples). • Confidence intervals are narrow when the sum of squared errors is small, the sample is large, or there’s a lot of variation in X. • How to present results from a regression model.

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Translations of the oldest existing Buddhist writings are changing how scholars believe the religion developed. Nineteen-hundred-year-old manuscripts from the Gandhara region offer a rare glimpse of the religion as it expanded from its native India around the first century C.E. Written in the language called Gandhari, the texts are adding a new dimension to the Buddhist canons of ancient Sanskrit and the living traditions of Pali, Chinese, and Tibetan. A team of linguists and historians at the University of Washington are reconstructing these texts in an NEH-supported project with the British Library. What the translations reveal is akin to discovering that the world is round, at least in the world of Buddhist studies. Tradition purports that Buddhist texts form a single line of development from the Buddha: as the religion spread, the Buddha's teachings were heard by disciples, then retold word for word, generation by generation. "The Gandhari texts indicate that this is not quite accurate," explains Collette Cox, an expert on early Buddhism for the project. "We now have a multiplicity of different versions of the teachings from a very early period, which indicates that there may have been much more acceptance among Buddhists of presenting the teachings in their own way. "It's like Greek philosophy if we only knew Plato through Socrates, and we didn't know any other Greek philosophers," continues Cox. "This is filling in a lot of gaps and adding complexity, making our tasks as scholars much more difficult, but also making our scholarly work much closer to what may have actually happened historically." Buddhism originated in India around the fifth century B.C.E., when Siddhartha, the Buddha, developed his belief system. Although Buddhism in India had essentially disappeared by 1200 C.E., varying strains of the faith survive today in Sri Lanka, South Korea, Nepal, and other Asian countries. The big question, says Cox, is "how did it get from point A to point B?" Scholars have long speculated Gandhara, located in northern Pakistan and eastern Afghanistan, to be the area from which Buddhism flowed into east and central Asia beginning around 100 C.E. Temples and shrines date back to the period and the region was near the Silk Road, providing the opportunity to travel and proselytize. But this theory was unsubstantiated because there was scant written evidence of Buddhists in the area. "If you look at the map of Asia and study the trade routes and the history and so forth, you see the likelihood that Gandhara was the funneling point from which Buddhism came out of India, went up into central Asia," says Richard Salomon, the project's director and head linguist. "And this time of the earliest manuscripts--first, second century C.E.--is when you first start seeing the presence of Buddhism in China. We think Buddhist monks went to China as missionaries, and probably went through Gandhara and carried manuscripts very similar to ones that we're looking at now." This long-standing theory is closer to being proven because of these new manuscripts, says Cox. "Even though this hypothesis has been around for a while, we had absolutely no information about what Buddhism was like in that area. Until recently, we had archaeological remains and inscriptions but only one text. It was sort of like a black hole. . . . These documents actually connect Indian Buddhism to East Asian Buddhism." Similarities are found in titles and content, and some Gandhari words appear in a transformed fashion in Chinese. "Sometimes names are translated literally in the Chinese texts," says Cox, "but at other times they are translated using Chinese characters to mimic the pronunciation of the original language. In the cases of these translations, these transliterations suggest the original texts were written in the Indic dialect of Gandhari." Unlike religions such as Islam or Judaism, in which language is critical to the scripture, the Buddha said that monks should speak to the people in their own language. "We think that as Buddhism spread across India and beyond, the missionaries would pick up the local language. Presumably, Buddhist communities would write down the texts in their own language. So we can imagine there were versions in other local dialects that haven't survived," Salomon says. The only previous Gandhari text was discovered in central Asia in 1892 and published by John Brough in the 1960s. No other manuscripts surfaced until 1994 when the British Library bought a collection of twenty-nine Gandhari scrolls with a mysterious history. "The British Library purchased the scrolls at auction in the U.K. with no firm information about their provenance," says Salomon, who doubts we will ever know exactly where they were found. He places them in Gaandhara because of several reports from the past two hundred years of the existence of similar documents, and of hundreds of reliquaries inscribed in Gandhari that have been unearthed near Buddhist monasteries in the region. The manuscripts were buried in spherical clay pots. "Most likely the Buddhist monks dropped the manuscripts in these," says Salomon, "then they'd put a plate or dish on the pot's neck and maybe seal it with wax." The pots were buried in stupas, large mounds or stone structures that, among other things, hold sacred objects. But there was one major flaw: water seeped into the pots. Documents stored along the edges rotted, while those that were inside, on top, or in the middle were better preserved. The documents were rolled and folded before they were stored-Salomon likens the ancient scrolls to squashed cigars. Before unrolling a document for translation, British conservators had to moisturize it by placing it in a humidor. The first book published by the project, Ancient Buddhist Scrolls from Gandhara, explains the process: "Due to the inherent fragility of the birch bark with its horizontal striations, each layer of the scroll tended to form a separate fragment of varying length as it was unrolled, but it was possible to preserve in very large measure the original sequence of layers and their texts. These fragments were then encapsulated between layers of glass. . . ." The resulting manuscripts vary wildly in size and condition, ranging from a legible two-foot by eight-inch section to manuscripts that have broken into hundreds of pieces. The library shoots high-detail photographs of the fragments, then scans them into a computer. "About 80 to 90 percent of what we do is on computer," Salomon estimates. The University of Washington group reconstructs the manuscripts on-screen, moving the pieces around like jigsaw puzzles with decomposed and missing pieces. "Some of the documents are in terrible shape," he says. "I was just looking at one of the fragments published a couple years ago, and there are 105 pieces of it." The British Library enlisted Salomon, one of only a handful of people familiarwith Gandhari, to translate the scrolls. Apart from the 1892 text, the language was known in fragmented fashion through coins, legal and administrative documents, and inscriptions on vessels that held sacred objects, such the bones of the Buddha. The reliquaries "often have dedicatory or ritual inscriptions on the outside, and they were usually in a basic formula: So-and-so, the son of so-and-so and wife of so-and-so, dedicates these relics in such-and-such a place," Salomon says. "There's not much variation, so you get a small body of vocabulary." But with the new documents the language has begun to take shape. "It's now much richer and more diverse. There's a huge amount of new vocabulary and different styles of writing." The writing sometimes bears striking resemblances to other known canons, such as in the first verse of the Rhinoceros Sutra, an early text found in several languages advocating solitary asceticism as a path towards enlightenment. Although the manuscript is fragmented, Salomon and his team have used the Pali text in order to fill in missing words and to provide the final line found in the other canons of Buddhism: "laying aside violence toward all beings, not harming even one among them, benevolent and sympathetic/ with a loving mind, one should wander alone/ like the rhinoceros." The materials are classified in three basic categories: sutras, which are teachings of the Buddha; avadanas, or pious legends, which concern the Buddha or his disciples; and scholastic materials which contain debates over doctrine. The Gandhari manuscripts are distinguishable from the other canons in their ordering of material, the multiple hands transcribing the documents, and what seems to be the monks' willingness to alter the texts. Several avadanas are particular to northwest India and mention local political figures. "From that we might speculate that every region had its own collection of these stories," says Cox. "These characters do not appear in the past life stories of the Pali or Chinese texts, so it suggests we have a local composition. It seems to be an attempt to make them locally relevant." "The material as a whole contains large amounts of both general Buddhist literature and local materials," says Salomon. "In the long run, it's going to show us the relationship between common, pan-Buddhist tradition and local, individual tradition." In addition there appears in the Gandhari texts a previously unknown school of the faith. Cox has studied one doctrine in the collection written from a perspective different from any other that exists in doctrinal treatises. The opponent in this text is familiar. "This opponent is called Sarvastivada, or 'those who claim that everything exists.' If we look at Chinese translations, we have a number of Sarvastivada doctrinal texts that represent the position of the Saravastivadans in which they're arguing against other people. But we don't have any texts in which they're the objects of criticism. Here we have a text that's clearly written from the opponent of the Saravastivadans." Cox believes that many schools composed doctrinal treatises with their own variations, but only a few remain. "We assumed those were the dominant schools and therefore the dominant position, and that they formed the basis for later Buddhist doctrinal interpretation in India and in East Asia and so forth. This may not be true. "This provides more proof that there was greater variety in the doctrinal, scholastic world than we knew about previously," says Cox. "In the Christian tradition, the Gnostic texts impact our knowledge of early Christian documents," she continues. "The Gandhari texts provide similar evidence on the Buddhist side, forcing us to examine assumptions based on only the other sets of early Buddhist documents. We're no longer looking at Buddhists as transmitting a single body of material that came to be different over time."

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Hodgkin's disease, also called Hodgkin's lymphoma, is a type of cancer involving tissues of the lymphatic system, or lymph nodes. Its cause is unknown, although some interaction between individual genetic makeup, family history, environmental exposures, and infectious agents is suspected. Hodgkin's lymphoma can occur at any age, although the majority of these lymphomas occur in people age 15-34, and over the age of 60. Lymphoma is a cancer of the lymphatic system. Depending on the specific type, a lymphoma can have any or all of the characteristics of cancer: rapid multiplication of cells, abnormal cell types, loss of normal arrangement of cells with respect to each other, and invasive ability. Causes & symptoms Hodgkin's lymphoma usually begins in a lymph node. The node enlarges and--similar to enlarged lymph nodes due to other infectious causes--may or may not cause any pain. Hodgkin's lymphoma progresses in a fairly predictable way, traveling from one group of lymph nodes to another, unless it is treated. More advanced cases of Hodgkin's involve the spleen, liver, and bone marrow. Constitutional symptoms--symptoms that affect the whole body--are common. They include fever, weight loss, heavy sweating at night, and itching. Some patients note pain after drinking alcoholic beverages. As nodes swell, they may push against nearby structures, resulting in other local symptoms. These symptoms include pain from pressure on nerve roots, as well as loss of function of specific muscle groups served by the compressed nerves. Kidney failure may result from compression of the ureters, the tubes which carry urine from the kidneys to the bladder. The face, neck, or arms may swell due to pressure slowing the flow in veins that should drain blood from those regions (superior vena cava syndrome). Pressure on the spinal cord can result in leg paralysis. Compression of the trachea and/or bronchi (airways) can cause wheezing and shortness of breath. Masses in the liver can cause the accumulation of certain chemicals in the blood, resulting in jaundice--a yellowish discoloration of the skin and the whites of the eyes. As Hodgkin's lymphoma progresses, a patient's immune system becomes less and less effective at fighting infection. Thus, patients with Hodgkin's lymphoma become increasingly more susceptible to both common infections caused by bacteria and unusual (opportunistic) infections caused by viruses, fungi, and protozoa. Diagnosis of Hodgkin's lymphoma requires the removal of a sample of a suspicious lymph node (biopsy) and careful examination of the tissue under a microscope. In Hodgkin's lymphoma, certain characteristic cells--Reed-Sternberg cells--must be present in order to confirm the diagnosis. These cells usually contain two or more nuclei--oval, centrally-located structures within cells which houses their genetic material. In addition to the identification of these Reed-Sternberg cells, other cells in the affected tissue sample are examined. The characteristics of these other cells help to classify the specific subtype of Hodgkin's lymphoma. Once Hodgkin's disease has been diagnosed, staging is the next important step. This involves computed tomography scans (CT scans) of the abdomen, chest, and pelvis, to identify areas of lymph node involvement. In rare cases, a patient must undergo abdominal surgery so that lymph nodes in the abdominal area can be biopsied (staging laparotomy). Some patients have their spleens removed during this surgery, both to help with staging and to remove a focus of the disease. Bone marrow biopsy is also required unless there is obvious evidence of vital organ involvement. Some physicians also order a lymphangiogram--a radiograph of the lymphatic vessels. Staging is important because it helps to determine what kind of treatment a patient should receive. On one hand, it is important to understand the stage of the disease so that the treatment chosen is sufficiently strong to provide the patient with a cure. On the other hand, all the available treatments have serious side effects, so staging allows the patient to have the type of treatment necessary to achieve a cure, and to minimize the severity of short and long-term side effects from which the patient may suffer. Hodgkin's disease is a life-threatening disease, and a correct diagnosis and appropriate treatment with surgery, chemotherapy, and/or radiation therapy is critical to controlling the illness. Acupuncture, hypnotherapy, and guided imagery may be useful tools in treating pain symptoms associated with Hodgkin's. Acupuncture involves the placement of a series of thin needles into the skin at targeted locations on the body known as acupoints in order to harmonize the energy flow within the human body. In guided imagery, the patient creates pleasant and comfortable mental images that promote relaxation and improve a patient's ability to cope with discomfort and pain symptoms. Other guided imagery techniques involve creating a visual mental image of pain. Once the pain can be visualized, the patient can adjust the image to make it more pleasing, and thus more manageable, to them. A number of herbal remedies are also available to lessen pain symptoms and promote relaxation and healing. However, individuals should consult with their healthcare professionals before taking them. Depending on the preparation and the type of herb, these remedies may interact with or enhance the effects of other prescribed medications. Treatment of Hodgkin's lymphoma has become increasingly effective over the years. The type of treatment used for Hodgkin's depends on the information obtained by staging, and may include chemotherapy (treatment with a combination of drugs), and/or radiotherapy (treatment with radiation to kill cancer cells). Both chemotherapy and radiotherapy have unfortunate side effects. Chemotherapy can result in nausea, vomiting, hair loss , and increased susceptibility to infection. Radiotherapy can cause sore throat , difficulty in swallowing, diarrhea, and growth abnormalities in children. Both forms of treatment, especially in combination, can result in sterility (the permanent inability to have offspring), as well as heart and lung damage. Hodgkin's is one of the most curable forms of cancer. Current treatments are quite effective, especially with early diagnosis. Children have a particularly high rate of cure from the disease, with about 75% still living cancer-free 20 years after their original diagnosis. Adults with the most severe form of the disease have about a 50% cure rate. - The removal of a small sample of tissue, in order to carefully examine it under a microscope. This helps in the diagnosis of cancer, and can also reveal infection or inflammation. - Treatment for a disease involving various chemical or drug preparations; this term tends to refer to treatment for forms of cancer in particular. - Involving the whole body. A constitutional symptom, for example, is one that is not focused entirely in the diseased organ system, but affects the whole system (such as fever). - Treatment for a disease involving carefully measured exposure to radiation. - Using various methods of diagnosis to determine the extent of disease present in an individual. Staging is important as a way of determining the appropriate type of treatment for a particular disease, as well as helping to predict an individual's chance for cure from a particular disease. For Your Information - Dollinger, Malin, et al. Everyone's Guide to Cancer Therapy. Kansas City: Andrews McMeel Publishing, 1997. - Freedman, Arnold S. and Lee M. Nadler. "Hodgkin's Disease." In Harrison's Principles of Internal Medicine, edited by Anthony S. Fauci, et al. New York: McGraw-Hill, 1998. - The Lymphoma Research Foundation of America, Inc. 8800 Venice Boulevard, Suite 207, Los Angeles, CA 90034. (310) 204-7040. http://www.lymphoma.org. Gale Encyclopedia of Alternative Medicine. Gale Group, 2001.

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# Polynomial Calculator ## By James Park and Woocheol Hyun Back to main page This calculator was made as an extra credit math project by James Park and Woocheol Hyun. As it currently is, the calculator can determine all rational roots and irrational roots within x: [-100, 100]. In addition, the calculator can determine y-intercepts, whether the function is odd or even, the first and second derivatives, and local extrema. ### How the function is broken down When an equation is typed into the input box, the program breaks up the polynomial into terms. For instance, 2x^2+3x-1 would be broken down into 2x^2, 3x, and -1. Next, each separate term is broken down into a coefficient and exponent, which are stored in arrays. Using this information is is then possible to find our desired data. ### How it finds the roots All rational roots are found using the rational root theorem. Our algorithm finds all possible rational roots and then plugs them into the function until a root is found. Irrational roots are found using Newton's method for approximating roots. A good explanation can be found here. ### How it finds derivatives Once the program has arrays with the coefficients and exponents the program uses the power rule to find the derivative of the function. ### How it finds y-intercepts To find the y-intercept of a function the program goes through the array of exponents to find the term with an exponent of 0 (x^0). It then takes the corresponding coefficient. This number is the constant term and will be the y-intercept. If no such term exists, the y-intercept is 0. ### Even or odd The program sorts the arrays of exponents and coefficients from largest exponent to smallest. Then the calculator finds if the first (largest) exponent is even or odd. ### How it finds the local extrema Local maximums and minimums are calculated through the derivative of the function. By finding the zeros of the derivatives the program can make a list of possible maxes and mins. Then using signs analysis the program is able to finalize the list of extrema. The program is limited to finding the min/max within the domain x: [-100, 100].

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# Calculus Find the limit lim tan^2(7x)/9x x-> 0 1. 👍 0 2. 👎 0 3. 👁 137 1. recall that tanx/x = sinx/x = 1 as x -> 0 so, this limit is 7/9 * tan(7x)/(7x) * tan(7x) -> 7/9 * 1 * 0 = 0 1. 👍 0 2. 👎 0 ## Similar Questions 1. ### CALCULUS - need help! Determine the limit of the trigonometric function (if it exists). 1. lim sin x / 5x (x -> 0) 2. lim tan^2x / x (x ->0) 3. lim cos x tan x / x (x -> 0) asked by ac on October 1, 2007 2. ### L'Hopitals Rule 5) Use the L’Hopital’s method to evaluate the following limits. In each case, indicate what type of limit it is ( 0/0, ∞/∞, or 0∙∞) lim x→2 sin(x^2−4)/(x−2) = lim x→+∞ ln(x−3)/(x−5) = lim x→pi/4 asked by Jane on December 7, 2014 3. ### calc need to find: lim as x -> 0 of 4(e^2x - 1) / (e^x -1) Try splitting the limit for the numerator and denominator lim lim x->0 4(e^2x-1) (4)x->0 (e^2x-1) ______________ = ________________ lim lim x->0 e^X-1 x->0 e^x-1 Next solve for asked by brian on September 26, 2006 4. ### Calculus Limits Question: If lim(f(x)/x)=-5 as x approaches 0, then lim(x^2(f(-1/x^2))) as x approaches infinity is equal to (a) 5 (b) -5 (c) -infinity (d) 1/5 (e) none of these The answer key says (a) 5. So this is what I know: Since asked by Anonymous on March 2, 2019 5. ### Calc I have a test soon.. and I really need to know how to do this problem.. please help!!! lim as x-->0 sin^2(x)/tan(x^2) the answer is 1, but I have no clue how to get that! Use series expansions. Look up Taylor expansion on google asked by Urgent on March 30, 2007 1. ### Math-Limits sqrt(1+tan x)-sqrt(1+sin x) lim all divided by x^3 x-->0 Use that Sqrt[1+x] = 1+ 1/2 x + 1/2 (-1/2)/2 x^2 + 1/2(-1/2)(-3/2)/6 x^3 + O(x^4) You can thus write the numerator as: 1/2 [tan(x) - sin(x)] - 1/8 [tan^2(x) - sin^2(x)] + asked by Matt on March 5, 2007 2. ### calculus Lim Tan^2(3x)/2x^2 X-0 Find the Limit? asked by Roshan on September 8, 2011 3. ### Calculus-Limits Okay, i posted this question yesterday, however, I did not really understand the answer I received. If your the one who answered my question, could you please elaborate. If not, could you try to answer this tough, for me, asked by Matt on March 6, 2007 4. ### limiting position of the particle A particle moves along the x axis so that its position at any time t>= 0 is given by x = arctan t What is the limiting position of the particle as t approaches infinity? Answer is pi/2 How do I solve this? Thanks a lot. You want asked by Jen on October 20, 2006 5. ### calculus Find the limit if it exist lim t-->pi/3 (tan(t)/t)+1 asked by erica on September 6, 2012 More Similar Questions

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I had £3 Riddle answer, I had 3 Pound riddle: Today we are going to solve I had £3 My mom gave me £10 riddle. It has been a few days since a lot of people are showing their interest in this riddle and are also challenging their friends and family members to solve it. As per the data, more than 95% of people are unable to solve I had £3 Riddle and end up with the wrong answer. Here we are going to provide the correct I had £3 Riddle answer. So now let us get started. I had £3. My mom gave me £10 while my dad gave me £30. My aunt and uncle gave me £100. I had another £5. How much money did I  have? ## I had £3 Riddle Explanation As we now know that the I had £3 Riddle answer is £8. So now let us explain the solution to you. So to solve this riddle, let us have a look at the riddle question once again. The riddle says “I had £3. My mom gave me £10 while my dad gave me £30. My aunt and uncle gave me £100. I had another £5. How much money did I really have?“. If you read the riddle carefully, it says How much money did I really have?. Here the word “DID” refers to the past tense. So logically the question is how much money you had before you borrowed £10 from your mom, £30 from dad & £100 from aunt and uncle. In the riddle, it is clearly mentioned that you had £3 (in the first statement) and another £5 in the 4th statement. So total you had £3 + £5 = £8.

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# How to prove that sec²θ + csc²θ = sec²θ × csc²θ? • Difficulty Level : Basic • Last Updated : 03 Sep, 2021 Trigonometry is a study of the properties of triangles and trigonometric functions. It is used a lot in engineering, science, for building video games, and more. It deals with the relationship between ratios of the sides of a right-angled triangle with its angles. These ratios which are used for the study of this relationship are called Trigonometric ratios. right-angled triangle ### Trigonometric Ratios There are six basic trigonometric ratios that establish the co-relation between sides of a right triangle with the angle. If θ is  the angle formed between the base and hypotenuse, in a right-angled triangle (as shown in the above figure), then Attention reader! All those who say programming isn't for kids, just haven't met the right mentors yet. Join the  Demo Class for First Step to Coding Coursespecifically designed for students of class 8 to 12. The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future. sin(θ) = Perpendicular/Hypotenuse cos(θ) = Base/Hypotenuse tan(θ) = Perpendicular/Base The values of the other three function ,that is, cosec(θ), sec(θ), cot(θ) depend on sin(θ), cos(θ), tan(θ) respectively. cot (θ) = 1/tan (θ) = Base/Perpendicular sec (θ) = 1/cos (θ) = Hypotenuse/Base cosec (θ) = 1/sin (θ) = Hypotenuse/Perpendicular ### Trigonometric Identities A trigonometric equation that holds for every viable value for an input variable on which it is defined is called trigonometric identities. One identity that we are  familiar with is the Pythagorean Identity, #### Sin2θ + Cos2θ = 1 Divide both sides of the Pythagorean Identity by cosine squared, which is allowed, then, [cos2θ + sin2θ]/cos2θ = 1/cos2θ cos2θ/cos2θ + sin2θ/cos2θ = 1/cos2θ 1 + sin2θ/cos2θ = 1/cos2θ (using the definition tanθ = sinθ/cosθ) 1 + tan2θ = 1/cos2θ (using the definition secθ = 1/cosθ) 1 + tan2θ = sec2θ Therefore the next trigonometric identity is, 1 + tan2θ  = sec2θ Similarly, if divide both sides of the Pythagorean Identity by sine squared then we obtained the last identity, cot2θ + 1 =  cosec2θ ### How to prove that sec2θ + csc2θ = sec2θ × csc2θ? Proof: To solve above problem we require below specified trigonometric identities and ratios : sec(θ) = 1/cos(θ)  and  cosec(θ) = 1/sin(θ)  ⇢ Eq. 1 sin2θ + cos2θ =1  ⇢ Eq. 2 sec2θ = 1 + tan2θ  ⇢ Eq. 3 cosec2θ  = 1+ cot2θ  ⇢ Eq. 4 There are two ways to solve this problem 1. To prove LHS = RHS using identities LHS = sec2(θ) + cosec2(θ) =(1+ tan2θ) + (1+cosec2θ)   (from 3 and 4) =2 + tan2θ + cosec2θ RHS= sec2θ × cosec2θ =(1+tan2θ) × (1+cot2θ) =2 + tan2θ + cot2θ Therefore, LHS = RHS. 2. By using trigonometric ratios LHS= sec2θ + cosec2θ (from 1), [1/cos2θ ]  +  [1/sin2θ] =  [sin2θ + cos2θ] / [cos2θ × sin2θ] (from 2), 1/[cos2θ × sin2θ] = sec2θ × cosec2θ = RHS Hence, the given trigonometric equation can be solved in two ways as mentioned above Therefore, sec2(θ) + cosec2(θ) = sec2(θ) × cosec2(θ). ### Similar Problems Question 1: Prove, tan4(θ) + tan2(θ) = sec4(θ) – sec2 (θ)   [Hint: take tan2(θ) as common] Solution: LHS= tan4θ + tan2θ =  tan2θ (tan2θ + 1) = (sec2θ – 1) (tan2θ + 1)        {since, tan2θ = sec2θ – 1} =(sec2θ – 1) sec2θ           {since, tan2θ + 1 = sec2θ} =sec4(θ) – sec2(θ)  =  RHS Hence proved. Question 2: Prove, cos θ / [(1 – tan θ)] + sin θ / [(1 – cot θ)] = sin θ + cos θ Solution: LHS =  cos θ / [(1 – tan θ)] + sin θ / [(1 – cot θ)] =cos θ / [1 – (sin θ/cos θ)] + sin θ/[1 – (cos θ/sin θ)] = cos θ / [(cos θ – sin θ/cos θ] + sin θ / [(sin θ – cos θ/sin θ)] = cos2θ/(cos θ – sin θ) + sin2θ/(cos θ – sin θ) = (cos2θ – sin2θ)/(cos θ – sin θ) = [(cos θ + sin θ)(cos θ – sin θ)] / (cos θ – sin θ) = (cos θ + sin θ) =  RHS Hence proved. Question 3: Prove, (tan θ + sec θ – 1)/(tan θ – sec θ + 1) = (1 + sin θ) × sec θ Solution: LHS = (tan θ + sec θ – 1)/(tan θ – sec θ + 1) = [(tan θ + sec θ) – (sec2θ – tan2θ)]/(tan θ – sec θ + 1)       [ sec2θ – tan2θ = 1] = [ (tan θ + sec θ) – (sec θ + tan θ) (sec θ – tan θ)] / (tan θ – sec θ + 1) = [ (tan θ + sec θ) × (1 – sec θ + tan θ)] / (tan θ – sec θ + 1) = [(tan θ + sec θ) × (tan θ – sec θ + 1)] / (tan θ – sec θ + 1) = (tan θ + sec θ) = (sin θ/cos θ) + (1/cos θ) = (sin θ + 1) / cos θ = (1 + sin θ) × secθ  = RHS ; Hence proved. Question 4:  = cosec θ – cot θ     [Hint: Multiply numerator and denominator by (sec θ – 1)] Solution: LHS = (multiply numerator and denominator by (sec θ – 1)) =√[(sec θ -1)2 / tan2θ ]           {sec2θ = 1 + tan2θ ⇢ sec2θ – 1 = tan2θ} = (sec θ – 1) / tan θ = (sec θ/tan θ) – (1/tan θ) = [(1/cos θ) / (sin θ/cos θ)] – cot θ = [(1/cos θ) × (cos θ/sin θ)] – cot θ = (1/sin θ) – cot θ = cosec θ – cot θ = RHS, Hence proved. Question 5: Prove, (sin θ+cosec θ)2+(cos θ+sec θ)2=7+tan2(θ)+cot2(θ) Solution: LHS = {sin2θ + cosec2θ + 2 sinθ cosecθ } + {cos2θ+ sec2θ +2 cosθ secθ} ={sin2θ + cosec2θ + 2} + { cos2θ + sec2θ + 2} =sin2θ + cos2θ + sec2θ + cosec2θ + 4 =sec2θ + cosec2θ + 5                            [sin2θ +cos2θ = 1] = (1+ tan2θ) +  (1+ cot2θ) +5              [sec2θ = 1 + tan2θ ;  cosec2θ = 1+ cot2θ ] =7+ tan2(θ) + cot2(θ) = RHS Hence proved. My Personal Notes arrow_drop_up

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SummaryStudents investigate the effect that weight has on rocket flight. They construct a variety of drinking straw-launched rockets—"strawkets"—of different weights. Specifically, they observe what happens when the weight of a strawket is altered by reducing its physical size and using different construction materials. They also they determine the importance of weight distribution in rockets. In the continuing hypothetical story for this unit, what students learn about rocket weight adds to their background understanding in their effort to help Tess launch a communication satellite. When designing rockets, one important consideration is weight. The more a rocket weighs, the more energy or thrust is required to launch it. Engineers aim to make rockets as light as possible while still making them strong—and as inexpensively as possible. If a rocket structure is too light, it will not be strong enough to withstand the stress forces of a launch. Engineers sometimes must compromise between the rocket weight and the rocket materials cost. They also consider a rocket's the weight distribution to make sure it moves as intended. After this activity, students should be able to: - Discuss what affects the weight of a rocket. - Explain why the weight distribution of a rocket is important. - Identify some factors that engineers must consider when designing rockets. More Curriculum Like This Through the continuing storyline of the Rockets unit, this lesson looks more closely at Spaceman Rohan, Spacewoman Tess, their daughter Maya, and their challenges with getting to space, setting up satellites, and exploring uncharted waters via a canoe. Students are introduced to the ideas of thrust,... Students explore motion, rockets and rocket motion while assisting Spacewoman Tess, Spaceman Rohan and Maya in their explorations. First they learn some basic facts about vehicles, rockets and why we use them. Then, they discover that the motion of all objects—including the flight of a rocket and mo... Students discover the entire process that goes into designing rockets. They learn about many important aspects such as supplies, ethics, deadlines and budgets. They also learn about the engineering design process and that the first design is almost never the final design. Students revisit the Pop Rockets activity from Lesson 3, in which mini paper rockets are powered by the chemical reaction of antacid-tablets and water in plastic film canisters. This time, however, the design of their pop rockets is limited by budgets and supplies. They get a feel for the constraint... Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc. Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc. - Define a simple design problem reflecting a need or a want that includes specified criteria for success and constraints on materials, time, or cost. (Grades 3 - 5) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! - Generate and compare multiple possible solutions to a problem based on how well each is likely to meet the criteria and constraints of the problem. (Grades 3 - 5) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! - Plan and carry out fair tests in which variables are controlled and failure points are considered to identify aspects of a model or prototype that can be improved. (Grades 3 - 5) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! - Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. (Grade 3) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! - Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (Grade 4) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! - Models are used to communicate and test design ideas and processes. (Grades 3 - 5) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! - Test and evaluate the solutions for the design problem. (Grades 3 - 5) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! - Represent and interpret data. (Grade 3) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! - Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. (Grade 3) Details... View more aligned curriculum... Do you agree with this alignment? Thanks for your feedback! Each student needs: - 1 facial tissue - 1 cup; narrow cups work better but are not required - 4-inch length of cotton string - 1 half-sized piece of letter-sized paper, measuring 8.5 × 5.5 inches - 1 quarter-sized piece of letter-sized paper, measuring 4.25 × 5.5 inches - 1 pencil - 1 drinking straw - 2 cotton balls - 1 pair of scissors - Weight Analysis Worksheet 1 - Weight Analysis Worksheet 2 - Weight Quiz For the entire class to share: Are you able to jump into the air while holding a basketball? (Answer: Yes, you should be able to.) How about while holding a 16-pound bowling ball? (Answer: Maybe, but definitely not as high as you could with just the basketball.) What if you had a bowling ball in each hand and a book bag full of rocks on your back? (Answer: Most likely not!) Would Maya have an easier or harder time paddling her canoe if it were full of bowling balls? (Answer: It would be much harder to paddle!) What does this all mean? Well, simply put, the more weight you are holding, the harder it is to get off the ground. Rockets react the exact same way! Rockets are very heavy by themselves, and when we put more weight in them—such as astronauts and equipment, they become even heavier. As you might imagine, it is difficult for rockets to get off the ground. One important job of engineers is to counteract this problem by making rockets lighter. Do you think a rocket could ever be too light? How about too heavy? The weight of a rocket is incredibly important to engineers who design them, and especially to you as Tess' engineering team. More weight means more energy is required to get the rocket off the ground. Engineers strive to make rockets as light as possible while still making them strong, and all as inexpensively as possible. Engineers cannot simply just remove all the weight from a rocket because it needs to be able to carry fuel, electronics, cargo and a structure to hold it all together. And, in Tess' case, she needs to transport satellites up to space to communicate with Maya. If the rocket structure is too light, it will not be strong enough to withstand the stresses of the launch. Engineers could use super strong and light materials, such as titanium, but titanium is very expensive. This means engineers must consider the tradeoffs between weight and cost and come to some affordable yet safe compromise between the weight of the rocket and the cost of the rocket. Engineers also must be careful about which part of the rocket is heavier. They consider the weight distribution. Should they make the rocket heavier in the front, the back or equally heavy all around? Where does the cargo go? What might happen if the front of the rocket is much heavier than the back? Well, we will find out. Today we will attempt to answer these questions by making small paper rockets, called strawkets, and experimenting to see how weight affects their flight. center of gravity: The point at which the entire weight of a body may be thought of as centered so that if supported at this point, the body would balance perfectly. Before the Activity - Gather materials and make copies of the Weight Analysis Worksheet 1, Weight Analysis Worksheet 2 and Weight Quiz. - Print out the planet targets, Inner and Outer. If possible, do so in color and laminate for reuse. - Cut enough pieces of letter-sized paper into halves so that each student receives one half, measuring 8.5 × 5.5 inches. - Remove the straws from their paper packaging, if necessary. - Use tape or string to mark a starting line on the floor. - Lay out the planet targets on the floor beyond the starting line. For a somewhat realistic layout, use one of the attached patterns: Launch from the Earth or Launch from the Sun. Note: Refer to the Planet Comparison Datasheet for actual planet diameters and distances. With the Students - Have students make predictions, as described in the Assessment section. - Present to the class the Introduction/Motivation content. - Hand out materials. - Have students wrap one half-sheet of paper around a pencil, starting from the eraser end and working up to the graphite tip. When wrapping, spiral the paper to make a cone shape (see Figure 2); it helps to hold it tighter at the eraser end and wrap upward. - Have students tape the paper tube near each end so it keeps its shape. Then remove the pencil. Check the final length of paper tubing to make sure it is at least a few centimeters shorter than the straws; otherwise, students will have nothing to hold onto for the launch. If necessary, use scissors to cut the paper tube shorter. - Have students pinch and fold the smaller end of the tube over and tape it so it is airtight. This end is the "nose" of the strawket. See Figure 1. - Because engineers always consider safety measures in their designs, direct students to tape a cotton ball to the nose of each strawket. To prevent the cotton from falling off the strawket, place the tape over the top of the cotton ball (that is, not wrapped inside/out and placed underneath the cotton ball as it sits on the nose of the paper tubing). Note: Some cotton balls are big enough to pull apart; only use as much cotton as necessary to provide some protective padding. - Have students personalize their strawkets. Suggest they write their names or draw designs on them so they know which one is theirs. - Have students find the center of gravity (CG) of their strawkets by balancing them on the side of a finger. While they may not be able to balance it perfectly, they will be able to get an idea of where it is close to balancing. The spot touching the finger is the CG. Alternatively, students can fold a piece of paper in half to make a fulcrum on which to balance their strawkets. For strawkets with no fins or paper clips, expect the CG to be near the middle of the strawket (depending on how much cotton is used). - Have students sketch their strawkets on the worksheet 1, noting the location of the center of gravity on their sketches. - Have students measure the lengths of their strawkets and mark the exact middle. - Give each student a cup and a 4-inch piece of string. - Have students tie the piece of string onto the strawket at the middle mark. Start Landing Sequence! - Have students place the cup on the floor, open side up. Then, stand above the cup and hold the string attached to the strawket, centering their hand above the cup. Wait until the strawket stops swinging. Then drop the strawket toward the cup. - If the strawket did not land in the cup, have them add a paperclip to the tail end and try again. Once a paperclip end landing is achieved, have the students write on the worksheet the number of paperclips they used. - Have students mark the new center of gravity on the sketch and label it (using the method described earlier). - Blast Off: Have students launch their strawkets with the paperclips attached. Have each student launch from the Earth or Sun (depending on the pattern you selected before the activity). Direct students to insert their straws into their strawkets—holding onto the straw, not the paper part of their strawket—aim at a planet, and blow. Expect the straket to flip and land tail first. - Repeat steps 9-17, but place the paperclips on the nose this time, instead of the tail. Then, have them answer the worksheet questions. - On worksheet 2, have students write down whether they think a strawket made out of a tissue will work. (In general, tissue strawkets are too light! Not only does air resistance slow them down quickly, often after one or two launches, the tissue bunches up inside and the straw cannot be reinserted.) - Have students repeat steps 4-7 with a tissue this time. - Have students find the center of gravity of their strawkets, as before in step 9. - Have students sketch their strawkets on worksheet 2, and mark the center of gravity on their sketches. - Blast Off: Have each student launch from the Earth or Sun (depending on the pattern you selected before the activity). To do this, students insert their straws into their strawkets—holding onto the straw, not the tissue part of the strawket—aim at a planet, and blow. - After retrieving their strawkets, direct students to complete the worksheet 2 questions before launching a second time. Have them write down the factors that they think helped or hurt them. - Now, have students make mini-strawkets (see Figure 3). Using a quarter-sized piece of paper that is 4.25 × 5.5 inches in sizse, have them cut it as small as they like while warning them that making it too small will prevent them from being able to spiral it into a cone. - Have students repeat steps 1-9 with the mini piece of paper this time. - Have students sketch their rockets on worksheet 2 and mark the center of gravity on their sketches. - Blast Off: Have each student launch from the Earth or Sun (depending on the pattern you selected before the activity). To do this, students insert their straws into their strawkets—holding onto the straw, not the paper part of their strawket—aim at a planet, and blow. - Direct students to complete the worksheet 2 questions before launching a second time. Have them write down the factors they think helped or hurt them. - Conclude by administering a post-activity quiz, leading a class discussion and assigning some graphing practice using class data, as described in the Assessment section. Strawkets should not be launched while the previous student is retrieving her/his strawket. Specifically, strawkets should not be launched at another person. Make a few strawkets in advance to confirm that your materials are suitable. Also, it is a good idea to have some extra strawkets in case someone's gets lost or crushed during the activity. If you do not have access to enough pencils, use extra drinking straws instead to help wrap the paper cone. Distributing tape to each student can be difficult while demonstrating how to build. If possible, have have several helpers pass out the tape or have pieces stuck on the table edges in advance. The tape used to secure the cotton balls should be fairly long so they are adhered properly. Make sure students are not holding onto the strawket when they blow through the straw! Prediction: Have students predict which strawket will go the farthest (paper or tissue) and record a tallies of their predictions on the classroom board. Activity Embedded Assessment Worksheets: Have students use Weight Analysis Worksheet 1 and Weight Analysis Worksheet 2 to record measurements, follow along with the activity and answer questions. After they have completed their worksheets, have them compare answers with their peers. Quiz: Administer the Weight Quiz, which covers simple division as well as how thrust is affected by weight. Review the answers as a class. Discussion Questions: Solicit, integrate and summarize student responses. Ask the students the following questions: - With no paperclips on it, why is the center of gravity slightly toward the nose of a strawket? (Answer: Because the nose is made by folding the paper over and adding tape and a cotton ball, which adds weight to the nose.) - Is tissue a good material to use to build strawkets? (Answer: No, it is too light; it does not hold the cone shape and becomes unusable after several launches.) - Would a strawket made of lead would work very well? Could you blow hard enough to launch a lead strawket? (Answer: No and no.) Graphing Practice: Have students create bar graphs of the class results using the Results from Earth Math Sheet or the Results from Sun Math Sheet (depending on where the students started their rocket from in step 17). You can also have students make comparison bar graphs of the distances achieved for tissue strawkets and mini-strawkets using the same results sheets. Have students measure the distances their individual strawkets traveled and record each attempt. Have them graph the data to show that paper mini-strawkets fly farther than tissue strawkets. Have students attempt to make strawkets out of other materials such as low-grade paperboard, index cards or construction paper. Can students come up with any other design improvements (for example, more thrust, high pressure air blower, fixed launch position, etc.)? For kindergarten to third-grade students, you can accomplish the first part of this activity within one class period by only having students complete the first strawket design, steps 1-17. Have students count down to launch as well as the number of strawkets that make it to each planet. Make a bar graph to help them visualize the numbers. Have students draw picture of their strawkets, labeling the parts and materials. Ask them to explain how they think adding weight would affect the rocket distance. For older, fourth- and fifth-grade students, have them complete the entire activity and graph the class data. Have them tabulate their results using the Results from the Earth Math Sheet or Results from the Sun Math Sheet and work out a class average for each stage of the experiment. Have them determine how their new strawket designs affected the class average. James, Donald. NASA Quest, National Aeronautics and Space Administration. Teacher Information: Paper Rockets. Accessed February 15, 2006. http://quest.arc.nasa.gov/space/teachers/rockets/act5.html Vogt, Gregory. NASA Glenn Learning Technologies Project (LTP), Aerospace Education Services Project, Oklahoma State University, "Paper Rockets," edited by Roger Storm, NASA Glenn Research Center. Accessed January 25, 2006. http://www.grc.nasa.gov/WWW/K-12/TRC/Rockets/paper_rocket.html ContributorsJeff White; Brian Argrow; Luke Simmons; Jay Shah; Malinda Schaefer Zarske; Janet Yowell Copyright© 2006 by Regents of the University of Colorado Supporting ProgramIntegrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder The contents of this digital library curriculum were developed under grants from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education, and National Science Foundation (GK-12 grant no 0338326). However, these contents do not necessarily represent the policies of the Department of Education or National Science Foundation, and you should not assume endorsement by the federal government. Last modified: August 10, 2017

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Codes and Ciphers :: Vigenère Cipher In a Caesar Cipher, each letter of the alphabet is shifted along some number of places; for example, in a Caesar cipher of shift 3, A would become D, B would become E and so on. The Vigenere cipher consists of using several Caesar ciphers in sequence with different shift values. To encipher, a table of alphabets can be used, termed a tabula recta, Vigenère square, or Vigenère table. It consists of the alphabet written out 26 times in different rows, each alphabet shifted cyclically to the left compared to the previous alphabet, corresponding to the 26 possible Caesar ciphers. At different points in the encryption process, the cipher uses a different alphabet from one of the rows. The alphabet used at each point depends on a repeating keyword. For example, suppose that the plaintext to be encrypted is: The person sending the message chooses a keyword and repeats it until it matches the length of the plaintext, for example, the keyword "LEMON": Each letter is encoded by finding the intersection in the grid between the plaintext letter and keyword letter. For example, the first letter of the plaintext, A, is enciphered using the alphabet in row L, which is the first letter of the key. This is done by looking at the letter in row L and column A of the Vigenere square, namely L. Similarly, for the second letter of the plaintext, the second letter of the key is used; the letter at row E and column T is X. The rest of the plaintext is enciphered in a similar fashion: Plaintext: ATTACKATDAWN Key: LEMONLEMONLE Ciphertext: LXFOPVEFRNHR Decryption is performed by finding the position of the ciphertext letter in a row of the table, and then taking the label of the column in which it appears as the plaintext. For example, in row L, the ciphertext L appears in column A, which taken as the first plaintext letter. The second letter is decrypted by looking up X in row E of the table; it appears in column T, which is taken as the plaintext letter. A Gronsfeld cipher is identical to the Vigenere cipher with the exception that only 10 rows are used which allows the keyword to be a number instead of a word. A Beaufort cipher uses the same alphabet table as the Vigenère cipher, but with a different algorithm. To encode a letter you find the letter in the top row. Then trace down until you find the keyletter. Then trace over to the left most column to find the enciphered letter. To decipher a letter, you find the letter in the left column, trace over to the keyletter and then trace up to find the deciphered letter. An Autokey cipher is identical to the Vigenère cipher with the exception that instead of creating a keyword by repeating one word over and over, the keyword is constructed by appending the keyword to the begining of the actual plaintext message. A Running Key cipher is identical to the Vigenère cipher with the exception that the keyword is chosen to be a book or long passage.Source: Wikipedia Table of Contents

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Before the various land masses that are now Africa, India, Australia, and Antarctica went their separate ways some 200 million years ago, they were part of the supercontinent Gondwana. So, too, was a "lost continent" scientists say is now hiding beneath the island nation of Mauritius. In a video, Lewis Ashwal of South Africa's University of the Witwatersrand explains the oldest rock on Earth exists in continents, while younger rocks form the ocean basins. Formed with volcanic eruptions in the Indian Ocean, Mauritius is also made up of young rock less than 9 million years old, per a release. But remarkably, scientists have found that rock contains minerals that date back 3 billion years to "one of the earliest periods in Earth's history," per Live Science. These minerals known as zircons prove "there are much older crustal materials under Mauritius that could only have originated from a continent," says Ashwal, whose research is published in Nature. He believes Zircons found on a tiny fragment of Gondwana that separated from Madagascar were covered by lava during the volcanic eruptions that created Mauritius before the lava solidified into the Mauritius we see today, hiding the older materials beneath. This suggests "a complex splintering [of Gondwana] took place, with fragments of continental crust of variable sizes left adrift within the evolving Indian Ocean basin." A 2013 study found zircons in Mauritius' sand, but some argued the minerals might have been carried there on the wind or scientists' shoes. This study, however, is literally set in stone. (Mount St. Helens hides something, too.)

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What Is It? Hepatitis A is a viral infection that can inflame and damage the liver. Unlike other forms of hepatitis, hepatitis A is usually mild and does not last long. Usually spread in contaminated food or water, hepatitis A also can be passed during sexual practices that involve the anus. In rare cases, hepatitis A can be spread by contact with the blood of a person who has the infection, for instance, when intravenous drug users share needles. About 30% of people in the United States have been exposed to hepatitis A, but only a very small number of them develop symptoms from the disease. Americans most likely to get hepatitis A include: - People who eat shellfish taken from waters where raw sewage drains - Children and caregivers in daycare centers who are exposed to the stool of an infected child - International travelers If the infection is mild, there may not be any symptoms, especially in a child. When symptoms appear, they can include: - Loss of appetite - Tenderness in the stomach area - Dark, tea-colored urine - Yellowing of the eyes and skin (jaundice) Your doctor may ask whether you have eaten shellfish recently or traveled to a foreign country with poor sanitation. He or she will ask about your personal hygiene habits and whether you have been near someone who has hepatitis A. Your doctor will examine you to check for swelling and tenderness near your liver and for a yellowish color to your skin and the whites of your eyes. You will need to have blood tests to confirm the diagnosis. Hepatitis A usually lasts two to eight weeks, although some people can be ill for as long as six months. The infection is likely to last longer and be more severe in people who are older or are in poor health. You can reduce your risk of getting hepatitis A by following these basic guidelines: - Wash your hands thoroughly with soap after handling food, after using the bathroom and before eating. - Buy shellfish only at reputable food stores or restaurants. - If you catch your own shellfish, make sure that it comes from waters inspected regularly by health authorities. - If you are traveling to a developing country, avoid drinking water or eating food that may be contaminated, and get vaccinated for hepatitis A before your trip. - Avoid injecting illegal drugs. Outbreaks of hepatitis A have been seen among intravenous drug users. A vaccine to prevent hepatitis A should be routinely given to: - All children 1 year (12 through 23 months) of age - Anyone 1 year of age and older traveling to or working in countries with high or intermediate prevalence of hepatitis A (most of the developing countries) - Men who have sex with men - People with persistent liver disease, such as chronic hepatitis - People with HIV infection - People who require blood transfusions or products derived from donated blood (such as clotting factors for bleeding disorders) - Research workers who handle the hepatitis A virus in the laboratory. Children who are not vaccinated by 2 years of age can be vaccinated at later visits. For travelers, the vaccine series should be started at least one month before traveling to provide the best protection. If you have been exposed to someone with hepatitis A, your doctor may give you the hepatitis vaccine or an injection of hepatitis A immune globulin to help prevent you from getting symptoms of the illness. Sometimes both are given. You should contact your doctor as soon as you become aware of the exposure. After two weeks post exposure, the immune globulin shot is not effective. There are no drugs to treat hepatitis A. Doctors generally recommend getting bed rest, eating well-balanced meals, drinking plenty of fluids and avoiding alcoholic beverages. It is also essential to avoid medications that can be toxic to your liver, such as acetaminophen (Tylenol). When To Call a Professional Call your doctor if you suspect that you have been exposed to someone with hepatitis A or if you are showing symptoms of the illness. If you are planning to travel to a foreign country, ask your doctor whether you should be vaccinated against hepatitis A before your trip. Nearly everyone who gets hepatitis A will recover completely within a few weeks to months. A very small number of people can get severe disease. In very rare cases (less than one-tenth of 1% of patients), the disease can cause liver failure, which can result in death if a liver transplant cannot be arranged. In people who already had liver disease or other types of hepatitis, such as hepatitis B and hepatitis C, the risk of severe disease from hepatitis A is much higher. Centers for Disease Control and Prevention (CDC) 1600 Clifton Road Atlanta, GA 30333 National Digestive Diseases Information Clearinghouse (NDDIC) 2 Information Way Bethesda, MD 20892-3570

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The Perpetual Union is a feature of the Articles of Confederation and perpetual Union, which established the United States of America as a national entity. Under American constitutional law, this concept means that states are not permitted to withdraw from the Union. The concept of a Union of the American States originated gradually during the 1770s as the independence struggle unfolded. In his First Inaugural Address on Monday March 4, 1861, Abraham Lincoln stated: "The Union is much older than the Constitution. It was formed, in fact, by the Articles of Association in 1774. It was matured and continued by the Declaration of Independence in 1776. It was further matured, and the faith of all the then thirteen States expressly plighted and engaged that it should be perpetual, by the Articles of Confederation in 1778. And finally, in 1787, one of the declared objects for ordaining and establishing the Constitution was to form a more perfect Union." A significant step was taken on June 12, 1776, when the Second Continental Congress approved the drafting of the Articles of Confederation, following a similar approval to draft the Declaration of Independence on June 11. The purpose of the former document was not only to define the relationship among the new states but also to stipulate the permanent nature of the new union. Accordingly, Article XIII states that the Union "shall be perpetual". While the process to ratify the Articles began in 1777, the Union only became a legal entity in 1781 when all states had ratified the agreement. The Second Continental Congress approved the Articles for ratification by the sovereign States on November 15, 1777, which occurred during the period from July 1778 to March 1781. The 13th ratification by Maryland was delayed for several years due to conflict of interest with some other states, including the western land claims of Virginia. After Virginia passed a law on January 2, 1781 relinquishing the claims, the path forward was cleared. On February 2, 1781, the Maryland state legislature in Annapolis passed the Act to ratify and on March 1, 1781 the Maryland delegates to the Second Continental Congress in Philadelphia formally signed the agreement. Maryland's final ratification of the Articles of Confederation and perpetual Union established the requisite unanimous consent for the legal creation of the United States of America. The concept of a perpetual union appeared earlier in European political thought. In 1532, François the 1st signed the Treaty of Perpetual Union (fr. Traité d'Union Perpétuelle), which pledged the freedom and privileges of Brittany within the kingdom of France. In 1713, Charles de Saint-Pierre presented a plan “A project for settling an everlasting peace in Europe,” where in it is stated in Article 1 "There shall be from this day following a Society, a permanent and perpetual Union, between the Sovereigns subscribed." By itself the word perpetual appears much earlier in the history of political thought. In January 44 B.C., Denarii coins were struck with the image of Julius Caesar and the Latin inscription "Caesar Dic(tator in) Perpetuo". From the start the Union has carried with it importance in the national affairs. There was a sense of urgency in completing the legal Union during the American Revolutionary War. Maryland’s ratification act stated, “[I]t hath been said that the common enemy is encouraged by this State not acceding to the Confederation, to hope that the union of the sister states may be dissolved” The nature of the Union was hotly debated during a period lasting from the 1830s through the American Civil War. During the Civil War, the United States was called "the Union", which could be seen as highlighting what it was fighting for. When the United States Constitution replaced the Articles, nothing in it specifically stated that the Union is perpetual. Even after the Civil War, which had been fought by the North to prevent the South from leaving the Union, some still questioned whether any such inviolability survived after the Constitution replaced the Articles. This uncertainty also stems from the fact that the Constitution, technically an amendment of the perpetual Articles, was not ratified unanimously before going into effect, as required by the Articles (two states, North Carolina and Rhode Island, had not ratified and remained outside the Union when George Washington was sworn in as the first President). The United States Supreme Court ruled on the issue in the 1869 Texas v. White case. In that case, the court ruled that the drafters intended the perpetuity of the Union to survive: |“||By [the Articles of Confederation], the Union was solemnly declared to "be perpetual." And when these Articles were found to be inadequate to the exigencies of the country, the Constitution was ordained "to form a more perfect Union." It is difficult to convey the idea of indissoluble unity more clearly than by these words. What can be indissoluble if a perpetual Union, made more perfect, is not?||”| - "Abraham Lincoln's First Inaugural Address on March 4, 1861". AMDOCS: Documents for the Study of American History. Retrieved 2009-10-27. - "History of Brittany". Retrieved 2009-11-03. - Chris Brown,Terry Nardin and Nicholas J. Rengger, eds. (2002). International relations in political thought: texts from the ancient Greeks to the 1st World War. Cambridge University Press. - "Caesars coins". Retrieved 2011-06-13. - Papers of the Continental Congress, No. 70, folio 453 and No. 9, History of the Confederation - 74 U.S. 700 (1869)

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Chocolate used at $0.90 per pound is 20 pounds #### Explanation: Let the weight of chocolates worth$0.90 a pound be ${c}_{0.90}$ Let the weight of chocolates worth $1.50 a pound be ${c}_{1.50}$Initial conditions: c_(0.90)+c_(1.50)=30^("lb")" ".....................Equation(1) The value of the blend is ($1.10)/("pound") so the total value of the final blend is ($1.10)/cancel("pound")xx30 color(white)("d")cancel("pound") =$33.00 Thus we have: $0.90c_(0.90)+$1.5c_(1.5)=$33.00" "................Equation(2) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You may choose which one you so wish to substitute for. I choose ${c}_{0.90}$Consider $E q n \left(1\right)$Write as ${c}_{0.90} = 30 - {c}_{1.50} \text{ } \ldots \ldots E q u a t i o n \left({1}_{a}\right)$Using $E q n \left({1}_{a}\right)$substitute for $\textcolor{red}{{c}_{0.90}}$in $E q n \left(2\right)$giving: Dropping the units of measerment for now. $\textcolor{g r e e n}{0.90 \textcolor{red}{{c}_{0.90}} \textcolor{w h i t e}{\text{dddd.d}} + 1.5 {c}_{1.50} = 33.00}$$\textcolor{g r e e n}{0.90 \left(\textcolor{red}{30 - {c}_{1.50}}\right) + 1.5 {c}_{1.50} = 33.00}$$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{d")27-0.9c_(1.50)color(white)("d}} + 1.5 {c}_{1.50} = 33.00}$$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{d")27color(white)("ddddd")+0.6c_(1.50)color(white)("dddd}} = 33.00}$Subtract 26 from both sides color(green)(0.6c_(1.50)=6 Divide both sides by $0.6$${c}_{1.50} = 10 \leftarrow \text{ pounds}$${c}_{0.90} = 30 - 10 = 20 \text{ pounds}\$

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Вы находитесь на странице: 1из 7 # Simple Linear Regression Analysis I. Correlation Analysis Goal: measure the strength and direction of a linear association between two variables. Basic concepts Scatter diagram plot of individual pairs of observations on a two-dimensional graph; used to visualize the possible underlying linear relationship. Example: Consider the following hypothetical data: x y 1 2 3.5 4.5 2 2 4 3.5 3 3 7 6.5 3 8 3 4 6 7.9 4 5 6 7 9.4 9.3 6 6 7 7 11 10.5 12.4 11.5 7 10 8 15 8 8 11 13.7 ## The scatter plot is as follows: 16 14 12 10 8 6 4 y 2 0 0 2 4 x 6 8 10 Linear correlation coefficient () a measure of the strength of the linear relationship existing between two variables, say X and Y, that is independent of their respective scales of measurements. Some characteristics of : It can only assume values between -1 and 1. The sign describes the direction of the linear relationship between X and Y: If is positive, the line slopes upward to the right, i.e., as X increases, the value of Y also increases. If is negative, the line slopes downward to the right, and so as X increases, the value of Y decreases. If =0, then there is NO LINEAR RELATIONSHIP between X and Y. If is -1 or 1, there is perfect linear relationship between X and Y and all the points (x,y) fall on a straight line. A that is close to 1 or -1 indicates a strong linear relationship. A strong linear relationship does not necessarily imply that X causes Y or Y causes X. it is possible that a third variable may have caused the change in both X and Y, producing the observed relationship. The Pearson product moment correlation coefficient between X and Y, denoted by r, is defined as: =1 =1 =1 = 2 =1 2 =1 2 2 =1 =1 Example: Compute for r of the hypothetical data given above. Solution: x 1 2 2 2 3 3 3 3 4 4 5 6 6 6 7 7 7 8 8 8 95 y xy 3.5 3.5 4.5 9 4 8 3.5 7 7 21 6.5 19.5 8 24 6 18 7.9 31.6 7 28 9.4 47 9.3 55.8 11 66 10.5 63 12.4 86.8 11.5 80.5 10 70 15 120 11 88 13.7 109.6 171.7 956.3 x^2 y^2 1 12.25 4 20.25 4 16 4 12.25 9 49 9 42.25 9 64 9 36 16 62.41 16 49 25 88.36 36 86.49 36 121 36 110.25 49 153.76 49 132.25 49 100 64 225 64 121 64 187.69 553 1689.21 Sum ## We obtain the following values: n=20 =1 = =1 =95 =1 =171.7 956.3 2 =1 = 2 =1 = 553 1689.21 Substituting these values to the formula, we have: = 20 956.3 95(171.7) 20 553 952 (20 1689.21 (171.72 ) = 0.9511 Tests of Hypotheses for Null Hypothesis Ho =o Alternative Hypothesis Ha <o >o o = Test Statistic Critical Region (i.e., Reject Ho if) < ( = 2) > ( = 2) > /2 ( = 2) ( ) 2 1 2 Example: Consider the hypothetical data given above. Suppose that the linear correlation coefficient between X and Y in the past is 0.9. Determine if the correlation has significantly increased compared to the past. a. Ho: =0.90 b. =0.05 c. = d. = vs Ha: >0.90 ## ( ) 2 1 2 (0.9511 0.9) 18 10.95112 = 0.7019 e. Decision rule: Reject Ho if > = 2 = .05 18 = 1.734 f. Since t = 0.7019 is not greater than .05 18 = 1.734, we do not reject Ho. At 0.05 level of significance, there is a sufficient evidence to conclude that the correlation coefficient between X and Y is 0.9. NOTE: Even if two variables are highly correlated, it is not a sufficient proof of causation. One variable may cause the other or vice versa, or a third factor is involved, or a rare event may have occurred. II. ## Simple Linear Regression Analysis Goal: To evaluate the relative impact of a predictor on a particular outcome. The simple linear regression model is given by the equation = + 1 + Where - the value of the response variable for the ith element - the value of the explanatory variable for the ith element - regression coefficient that gives Y- intercept of the regression line. 1 - regression coefficient that gives the slope of the line - random error for the ith element, where are independent, normally distributed with mean 0 and variance 2 for i=1, 2, , n n number of elements Remark: The model tells us that two or more observations having the same value for X will not necessarily have the same value for Y. However, the different values of Y for a given value of X, say x i, will be generated by a normal distribution whose mean is + 1 , that is, = + 1 . This is known as the regression equation where the parameters and 1 are interpreted as follows: is the value of the mean of Y when X=0 1 is the amount of change in the mean of Y for every unit increase in the value of X. The random error It may be thought of as a representation of the effect of other factors, that is, apart from X, not explicitly stated in the model but do affect the response variable to some extent. Sources of random error: o Other response variables not explicitly stated in the model o Inherent and inevitable variation present in the response variable o Measurement errors Satisfies the following: o The error terms are independent from one another; o The error terms are normally distributed; o The error terms all have a mean of 0; and o The error terms have constant variance, 2 . Typical steps in doing a simple linear regression analysis: 1. Obtain the equation that best fits the data. 2. Evaluate the equation to determine the strength of the relationship for prediction and estimation. 3. Determine if the assumptions on the error terms are satisfied. 4. If the model fits the data adequately, use the equation for prediction and for describing the nature of the relationship between the variables. Obtaining the equation: Method of Least Squares The best-fitting line is selected as the one that minimizes the sum of squares of the deviations of the observed value of Y from its expected value. That is we want to estimate and 1 such that =1 2 is smallest, where = = + 1 Based on this criterion, the following formulas for b o , the estimate for , and b1 , the estimate for 1 , are obtained: = =1 =1 =1 2 =1 2 =1 = 1 Thus, the estimated regression equation is given by = + 1 Remarks: The estimated regression equation is appropriate only for the relevant range of X, i.e., for the values of X used in developing the regression model. If X=0 is not included in the range of the sample data, the will not have a meaningful interpretation. Example: Consider the given hypothetical example where we fit a linear model of the form = + 1 + Using the method of least squares, the following values are needed to estimate and 1 : n=20 =1 = =1 =95 =1 =171.7 956.3 2 =1 = 553 ## We get the values of bo and b1 as: 1 = 20 956.3 95 (171.7) = 1.383 20(553) 95 2 = 8.585 1.383 4.75 = 2.016 Hence, the prediction equation is given by: = 2.016 + 1.383 Interpretation: For every 1 unit increase in X, the mean of Y is estimated to increase by 1.383. Note that bo =2.016 has no meaningful interpretation since X=0 is not within the range of values used in the estimation. Mean Square Error The common variance of and Y, denoted by 2 , is given by: = 2 =1 2 2 where SSE stands for sum of squares due to error and MSE stands for mean square error. The MSE is the variance of the data, Y, about the estimated regression line, . Determining the strength of relationship between X and Y A (1-)100% Confidence Interval for 1 is (1 = 2 1 , 1 + = 2 1 ) 2 2 Where 1 = 2 =1 2 =1 ## A (1-)100% Confidence Interval for o is ( = 2 , + = 2 ) 2 2 ( 2 ) =1 2 2 =1 =1 Where = Test of Hypothesis concerning 1 Null Hypothesis Ho 1 =0 Alternative Hypothesis Ha 1 <0 1 >0 1 0 Test Statistic 1 1 Critical Region (i.e., Reject Ho if) < ( = 2) > ( = 2) > /2 ( = 2) Coefficient of Determination (R2 ) The proportion of the variability in the observed values of the response variable that can be explained by the explanatory variable through their linear relationship. The realized value of the coefficient of determination, r 2 , will be between 0 and 1. If a model has perfect predictability, then R2 =1; but if a model has no perfect predictive capability, then R2 =0. Interpretation: R2 *(100%) of the variability in the response variable, Y, can be explained by the explanatory variable, X, through the simple linear regression model. Residual (di) The difference between the observed value and predicted value of the response variable. That is, = . If indeed the variances of the error terms are constant, then the plot of the residuals versus X should tend to form a horizontal band, i.e., spread of the residuals should not increase or decrease with values of the independent variable.

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## Lines and also linear equations ### Graphs of lines Geometry taught us that precisely one line crosses through any two points. We deserve to use this fact in algebra together well. When drawing the graph the a line, we only require two points, and then use a right edge to attach them. Remember, though, that lines are infinitely long: they do not start and stop in ~ the two points we provided to attract them. Lines deserve to be expressed algebraically together an equation that relates the \$y\$-values to the \$x\$-values. We deserve to use the same fact that we used previously that 2 points are contained in exactly one line. With only two points, we can determine the equation of a line. Prior to we do this, let"s discuss some an extremely important characteristics of lines: slope, \$y\$-intercept, and also \$x\$-intercept. slope Think of the steep of a line as its "steepness": how conveniently it rises or drops from left to right. This worth is shown in the graph above as \$fracDelta yDelta x\$, i m sorry specifies how much the line rises or drops (change in \$y\$) together we relocate from left to appropriate (change in \$x\$). That is crucial to relate steep or steepness come the price of vertical change per horizontal change. A popular instance is that of speed, which measures the change in distance per adjust in time. Wherein a line deserve to represent the distance traveled at various points in time, the steep of the heat represents the speed. A steep heat represents high speed, conversely, very tiny steepness represents a lot slower rate of travel, or short speed. This is depicted in the graph below. Speed graph The upright axis represents distance, and also the horizontal axis represents time. The red line is steeper 보다 the blue and also green lines. Notification the distance traveled after one hour ~ above the red heat is around 5 miles. The is much higher than the street traveled on the blue or environment-friendly lines ~ one hour - about \$1\$ mile and also \$frac15\$, respectively. The steeper the line, the greater the street traveled every unit the time. In various other words, steepness or slope to represent speed. The red present is the fastest, with the greatest slope, and also the environment-friendly line is the slowest, with the smallest slope. Slope deserve to be classified in 4 ways: positive, negative, zero, and undefined slope. Hopeful slope means that as we relocate from left to ideal on the graph, the line rises. An unfavorable slope way that together we relocate from left to right on the graph, the line falls. Zero slope way that the line is horizontal: it no rises nor drops as we relocate from left come right. Vertical lines are claimed to have "undefined slope," together their slope appears to be part infinitely large, undefined value. Check out the graphs listed below that present each the the 4 slope types. positive slope: negative slope: Zero slope (Horizontal): undefined slope (Vertical): \$fracDelta yDelta x gt 0\$ \$fracDelta yDelta x lt 0\$ \$Delta y = 0\$, \$Delta x eq 0\$, for this reason \$fracDelta yDelta x = 0\$ \$Delta x = 0\$, so \$fracDelta yDelta x\$ is unknown Investigate the habits of a line by adjusting the steep via the "\$m\$-slider". Watch this video on steep for an ext insight right into the concept. \$y\$-Intercept The \$y\$-intercept that a heat is the suggest where the line crosses the \$y\$-axis. Note that this happens when \$x = 0\$. What space the \$y\$-intercepts the the currently in the graphs above? that looks like the \$y\$-intercepts room \$(0, 1)\$, \$(0, 0)\$, and also \$(0, 1)\$ because that the first three graphs. Over there is no \$y\$-intercept ~ above the fourth graph - the line never crosses the \$y\$-axis. Investigate the behavior of a heat by adjusting the \$y\$-intercept via the "\$b\$-slider". \$x\$-Intercept The \$x\$-intercept is a similar concept together \$y\$-intercept: that is the allude where the line crosses the \$x\$-axis. This happens as soon as \$y = 0\$. The \$x\$-intercept is not offered as often as \$y\$-intercept, together we will certainly see when determing the equation the a line. What are the \$x\$-intercepts the the currently in the graphs above? the looks prefer the \$x\$-intercepts space \$(-frac12, 0)\$ and \$(0, 0)\$ for the first two graphs. Over there is no \$x\$-intercept on the third graph. The fourth graph has an \$x\$-intercept at \$(-1, 0)\$. ### Equations of currently In order to write an equation that a line, us usually have to determine the steep of the heat first. Calculating slope Algebraically, steep is calculated as the proportion of the adjust in the \$y\$ value to the adjust in the \$x\$ worth between any kind of two clues on the line. If we have actually two points, \$(x_1, y_1)\$ and also \$(x_2, y_2)\$, steep is express as: \$\$box extslope = m = fracDelta yDelta x = fracy_2 - y_1x_2 - x_1.\$\$ note that we usage the letter \$m\$ to denote slope. A line the is an extremely steep has actually \$m\$ values through very huge magnitude, whereas together line that is not steep has \$m\$ values through very small magnitude. Because that example, slopes of \$100\$ and also \$-1,000\$ have much bigger magnitude than slopes of \$-0.1\$ or \$1\$. Example: uncover the slope of the line the passes through points \$(-2, 1)\$ and \$(5, 8)\$. making use of the formula because that slope, and also letting suggest \$(x_1, y_1) = (-2, 1)\$ and point \$(x_2, y_2) = (5, 8)\$, \$\$eginalign* m &= fracDelta yDelta x = fracy_2 - y_1x_2 - x_1\<1ex> &= frac8 - 15 - (-2)\<1ex> &= frac75 + 2\<1ex> &= frac77\<1ex> &= 1 endalign*\$\$ note that we chose allude \$(-2, 1)\$ as \$(x_1, y_1)\$ and suggest \$(5, 8)\$ together \$(x_2, y_2)\$. This to be by choice, together we might have let allude \$(5, 8)\$ it is in \$(x_1, y_1)\$ and point \$(-2, 1)\$ it is in \$(x_1, y_1)\$. Exactly how does that affect the calculate of slope? \$\$eginalign* m &= fracDelta yDelta x = fracy_2 - y_1x_2 - x_1\<1ex> &= frac1 - 8-2 - 5\<1ex> &= frac-7-7\<1ex> &= 1 endalign*\$\$ We watch the steep is the very same either method we pick the first and 2nd points. We deserve to now conclude the the slope of the line the passes through points \$(-2, 1)\$ and also \$(5, 8)\$ is \$1\$. Watch this video for an ext examples ~ above calculating slope. currently that we know what slope and \$y\$-intercepts are, we have the right to determine the equation of a heat given any type of two point out on the line. There room two primary ways to create the equation the a line: point-slope form and slope-intercept form. We will very first look in ~ point-slope form. Point-Slope kind The point-slope type of one equation the passes with the allude \$(x_1, y_1)\$ v slope \$m\$ is the following: \$\$box extPoint-Slope form: y - y_1 = m(x - x_1).\$\$ Example: What is the equation of the line has slope \$m = 2\$ and also passes through the allude \$(5, 4)\$ in point-slope form? utilizing the formula for the point-slope type of the equation the the line, we deserve to just substitute the steep and allude coordinate worths directly. In other words, \$m = 2\$ and \$(x_1, y_2) = (5, 4)\$. So, the equation of the heat is \$\$y - 4 = 2(x - 5).\$\$ Example: offered two points, \$(-3, -5)\$ and also \$(2, 5)\$, create the point-slope equation the the line that passes with them. First, us calculate the slope: \$\$eginalign* m &= fracy_2 - y_1x_2 - x_1\<1ex> &= frac5 - (-5)2 - (-3)\<1ex> &= frac105\<1ex> &= 2 endalign*\$\$ Graphically, we deserve to verify the slope by looking in ~ the readjust in \$y\$-values matches the readjust in \$x\$-values between the 2 points: Graph of line passing through \$(2, 5)\$ and \$(-3, -5)\$. You are watching: Equation of a vertical line in standard form We can now use one of the points along with the steep to create the equation that the line: \$\$eginalign* y - y_1 &= m(x - x_1) \ y - 5 &= 2(x - 2) quadcheckmark endalign*\$\$ us could additionally have supplied the other point to write the equation the the line: \$\$eginalign* y - y_1 &= m(x - x_1) \ y - (-5) &= 2(x - (-3)) \ y + 5 &= 2(x + 3) quadcheckmark endalign*\$\$ however wait! Those two equations look different. How have the right to they both describe the same line? If we simplify the equations, we watch that castle are certainly the same. Let"s do just that: \$\$eginalign* y - 5 &= 2(x - 2) \ y - 5 &= 2x - 4 \ y - 5 + 5 &= 2x - 4 + 5 \ y &= 2x + 1 quadcheckmark endalign*\$\$ \$\$eginalign* y + 5 &= 2(x + 3) \ y + 5 &= 2x + 6 \ y + 5 - 5 &= 2x + 6 - 5 \ y &= 2x + 1 quadcheckmark endalign*\$\$ So, using either allude to compose the point-slope form of the equation results in the very same "simplified" equation. We will see following that this simplified equation is another important type of straight equations. Slope-Intercept form Another method to express the equation of a heat is slope-intercept form. \$\$box extSlope-Intercept form: y = mx + b.\$\$ In this equation, \$m\$ again is the steep of the line, and also \$(0, b)\$ is the \$y\$-intercept. Choose point-slope form, every we need are 2 points in stimulate to write the equation the passes with them in slope-intercept form. Constants vs. Variables the is important to keep in mind that in the equation for slope-intercept form, the letter \$a\$ and also \$b\$ are constant values, together opposed come the letters \$x\$ and also \$y\$, which space variables. Remember, constants stand for a "fixed" number - it does no change. A variable can be one of many values - it have the right to change. A offered line includes many points, each of which has a unique \$x\$ and \$y\$ value, however that line only has one slope-intercept equation through one value each for \$m\$ and \$b\$. Example: offered the same two points above, \$(-3, -5)\$ and \$(2, 5)\$, compose the slope-intercept form of the equation that the line that passes v them. We already calculated the slope, \$m\$, over to be \$2\$. We can then use among the point out to fix for \$b\$. Making use of \$(2, 5)\$, \$\$eginalign* y &= 2x + b \ 5 &= 2(2) + b \ 5 &= 4 + b \ 1 &= b. endalign*\$\$ So, the equation of the heat in slope-intercept type is, \$\$y = 2x + 1.\$\$ The \$y\$-intercept that the line is \$(0, b) = (0, 1)\$. Look at the graph above to verify this is the \$y\$-intercept. In ~ what suggest does the line cross the \$y\$-axis? At an initial glance, it seems the point-slope and also slope-intercept equations of the line space different, but they really do describe the very same line. We have the right to verify this through "simplifying" the point-slope type as such: \$\$eginalign* y - 5 &= 2(x - 2) \ y - 5 &= 2x - 4 \ y - 5 + 5 &= 2x - 4 + 5 \ y &= 2x + 1 \ endalign*\$\$ Watch this video for much more examples on composing equations of present in slope-intercept form. ### Horizontal and Vertical lines currently that we have the right to write equations the lines, we require to consider two special instances of lines: horizontal and also vertical. Us claimed above that horizontal lines have slope \$m = 0\$, and that upright lines have undefined slope. How can we usage this to determine the equations of horizontal and vertical lines? upright lines Facts around vertical present If two points have the very same \$x\$-coordinates, only a upright line have the right to pass v both points. Each suggest on a vertical line has the same \$x\$-coordinate. If two points have actually the very same \$x\$-coordinate, \$c\$, the equation of the heat is \$x = c\$. The \$x\$-intercept of a vertical heat \$x = c\$ is the point \$(c, 0)\$. other than for the heat \$x = 0\$, vertical lines do not have a \$y\$-intercept. Example: think about two points, \$(2, 0)\$ and \$(2, 1)\$. What is the equation that the line the passes v them? Graph of heat passing with points \$(2, 0)\$ and also \$(2, 1)\$ First, keep in mind that the \$x\$-coordinate is the very same for both points. In fact, if us plot any allude from the line, we deserve to see that the \$x\$-coordinate will be \$2\$. We know that only a upright line have the right to pass through the points, therefore the equation of the line must be \$x = 2\$. But, how can we verify this algebraically? very first off, what is the slope? we calculate slope as \$\$eginalign* m &= frac1 - 02 - 2 \<1ex> &= frac10 \<1ex> &= extundefined endalign*\$\$ In this case, the slope worth is undefined, which renders it a upright line. Slope-intercept and also point-slope creates at this point, you could ask, "how have the right to I write the equation that a vertical line in slope-intercept or point-slope form?" The answer is that you really have the right to only compose the equation of a vertical heat one way. For vertical lines, \$x\$ is the same, or constant, for all values of \$y\$. Due to the fact that \$y\$ can be any kind of number for vertical lines, the change \$y\$ does not show up in the equation that a upright line. Horizontal currently Facts around horizontal present If 2 points have the very same \$y\$-coordinates, only a horizontal line deserve to pass with both points. Each suggest on a horizontal line has the same \$y\$-coordinate. If 2 points have the very same \$y\$-coordinate, \$b\$, the equation that the heat is \$y = b\$. The \$y\$-intercept the a horizontal heat \$y = b\$ is the point \$(0, b)\$. other than for the line \$y = 0\$, horizontal lines do not have actually an \$x\$-intercept. Example: take into consideration two points, \$(3, 4)\$ and \$(0, 4)\$. What is the equation that the line the passes v them? Graph of heat passing through points \$(3, 4)\$ and also \$(0, 4)\$ First, keep in mind that the \$y\$-coordinate is the very same for both points. In fact, if we plot any suggest on the line, we can see the the \$y\$-coordinate is \$4\$. We understand that only a horizontal line can pass v the points, therefore the equation of the line should be \$y = 4\$. How can we verify this algebraically? First, calculate the slope: \$\$eginalign* m &= frac4 - 40 - 3 \<1ex> &= frac0-3 \<1ex> &= 0 endalign*\$\$ Then, making use of slope-intercept form, we can substitute \$0\$ because that \$m\$, and also solve because that \$y\$: \$\$eginalign* y &= (0)x + b \<1ex> &= b endalign*\$\$ This tells united state that every point on the line has \$y\$-coordinate \$b.\$ since we know two clues on the line have \$y\$-coordinate \$4\$, \$b\$ have to be \$4\$, and so the equation that the line is \$y = 4\$. Slope-intercept and also Point-slope develops similar to vertical lines, the equation that a horizontal line deserve to only be written one way. For horizontal lines, \$y\$ is the very same for all worths of \$x\$. Because \$x\$ can be any number because that horizontal lines, the change \$x\$ walk not show up in the equation the a horizontal line. ### Parallel and Perpendicular currently now that we know exactly how to characterize lines by their slope, we have the right to identify if 2 lines space parallel or perpendicular by their slopes. Parallel currently In geometry, we space told the two distinct lines that execute not intersect are parallel. Looking in ~ the graph below, there space two lines the seem to never to intersect. What have the right to we say around their slopes? It shows up that the lines above have the exact same slope, and that is correct. Non-vertical parallel lines have actually the same slope. Any type of two vertical lines, however, are also parallel. That is essential to note that vertical lines have actually undefined slope. Perpendicular present We recognize from geometry the perpendicular lines form an edge of \$90^circ\$. The blue and also red currently in the graph below are perpendicular. What execute we an alert about their slopes? also though this is one particular example, the relationship in between the slopes applies to all perpendicular lines. Skipping the indications for now, notification the vertical adjust in the blue line amounts to the horizontal change in the red line. Likewise, the the vertical change in the red line amounts to the horizontal adjust in the blue line. So, then, what are the slopes that these 2 lines? \$\$ extslope of blue line = frac-21 = -2\$\$ \$\$ extslope of red line = frac12\$\$ The other truth to notification is that the indications of the slopes that the lines are not the same. The blue line has a negative slope and the red line has actually a optimistic slope. If us multiply the slopes, us get, \$\$-2 imes frac12 = -1.\$\$ This station and an adverse relationship in between slopes is true for all perpendicular lines, other than horizontal and vertical lines. below is another example of two perpendicular lines: \$\$ extslope that blue line = frac-23\$\$ \$\$ extslope of red line = frac32\$\$ \$\$ extProduct the slopes = frac-23 cdot frac32 = -1\$\$ Again, we see that the slopes of 2 perpendicular lines are an adverse reciprocals, and therefore, your product is \$-1\$. Recall that the reciprocal of a number is \$1\$ separated by the number. Let"s verify this with the examples above: The an adverse reciprocal that \$-2\$ is \$-frac1-2 = frac12 checkmark\$. The negative reciprocal of \$frac12\$ is \$-frac1frac12 = -2 checkmark\$. The negative reciprocal the \$-frac23\$ is \$-frac1-frac23 = frac32 checkmark\$. The an adverse reciprocal that \$frac32\$ is \$-frac1frac32 = -frac23 checkmark\$. 2 lines space perpendicular if one of the following is true: The product of their slopes is \$-1\$. One line is vertical and the various other is horizontal. ### Exercises Calculate the slope of the heat passing with the given points. 1. \$(2, 1)\$ and \$(6, 9)\$ 2. \$(-4, -2)\$ and also \$(2, -3)\$ 3. \$(3, 0)\$ and also \$(6, 2)\$ 4. \$(0, 9)\$ and also \$(4, 7)\$ 5. \$(-2, frac12)\$ and \$(-5, frac12)\$ 6. \$(-5, -1)\$ and \$(2, 3)\$ 7. \$(-10, 3)\$ and also \$(-10, 4)\$ 8. \$(-6, -4)\$ and also \$(6, 5)\$ 9. \$(5, -2)\$ and \$(-4, -2)\$ Find the steep of each of the adhering to lines. 10. \$y - 2 = frac12(x - 2)\$ 11. \$y + 1 = x - 4\$ 12. \$y - frac23 = 4(x + 7)\$ 13. \$y = -(x + 2)\$ 14. \$2x + 3y = 6\$ 15. \$y = -2x\$ 16. \$y = x\$ 17. \$y = 4\$ 18. \$x = -2\$ 19. \$x = 0\$ 20. \$y = -1\$ 21. \$y = 0\$ Write the point-slope kind of the equation of the line with the offered slope and also containing the given point. 22. \$m = 6\$; \$(2, 7)\$ 23. \$m = frac35\$; \$(9, 2)\$ 24. \$m = -5\$; \$(6, 2)\$ 25. \$m = -2\$; \$(-4, -1)\$ 26. \$m = 1\$; \$(-2, -8)\$ 27. \$m = -1\$; \$(-3, 6)\$ 28. \$m = frac43\$; \$(7, -1)\$ 29. \$m = frac72\$; \$(-3, 4)\$ 30. \$m = -1\$; \$(-1, -1)\$ Write the point-slope kind of the equation that the line passing with the provided pair that points. 31. \$(1, 5)\$ and also \$(4, 2)\$ 32. \$(3, 7)\$ and also \$(4, 8)\$ 33. \$(-3, 1)\$ and \$(3, 5)\$ 34. \$(-2, 3)\$ and also \$(3, 5)\$ 35. \$(5, 0)\$ and \$(0, -2)\$ 36. \$(-2, 0)\$ and \$(0, 3)\$ 37. \$(0, 0)\$ and \$(-1, 1)\$ 38. \$(1, 1)\$ and \$(3, 1)\$ 39. \$(3, 2)\$ and also \$(3, -2)\$ Exercises 40-48: compose the slope-intercept kind of the equation the the line through the given slope and also containing the given suggest in exercises 22-30. See more: L Is Running 4 Miles A Day Good, Run 5 Miles A Day Exercises 49-57: compose the slope-intercept kind of the equation that the heat passing through the provided pair of clues in exercises 31-39.

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Wild Side: Monarchs are in trouble; give them milkweed --Dan Wilcox, outdoor columnist Monarchs are the iconic North American summer butterfly; beautiful insects with dark orange, black and white wings. They are the only butterfly with long-distance annual migrations. Hundreds of millions of monarchs migrate from eastern North America to high-elevation oyamel fir forests in the mountains of central Mexico to spend the winter. Monarchs from western North America migrate to a few locations in California coastal forests. Monarch females lay a single egg on the underside of a leaf on milkweed plants. A single female can mate repeatedly and lay up to several hundred eggs. The one millimeter-sized eggs hatch in about four days. Around here, monarch larvae (caterpillars) eat common milkweed, swamp milkweed and spreading dogbane (a close relative of milkweeds). The distinctive striped white, green and black larvae shed their skin four times as they grow over 10 to 14 days. Their bright markings help warn bird predators that they contain toxic alkaloids from the milkweed plants. The mature larvae spin a silk pad on the bottom of a milkweed leaf, shed their skin, attach themselves hanging from the silk pad, and form a well-camouflaged light green chrysalis with a hard shell. After 10 to 14 days in the pupal stage, fully-formed adult butterflies hatch out, mate, and the females begin laying eggs. Adults in the summer generations live for only two to five weeks due to the high cost of reproduction. The last generations of monarchs migrate south in the late summer and early fall to survive the long winter. Most monarchs originate more than 1500 miles from their overwintering sites. These adults can live up to eight or nine months, returning north in March and April to reproduce. Monarchs are in trouble. Their populations are declining markedly due to diminishing over-wintering and summer habitats. Deforestation in overwintering sites in Mexico has eliminated some sites and reduced shelter and water available at other sites. In the United States, chemical-intensive agriculture, conversion of pasture and CRP lands to row crops, urban development and roadside mowing have greatly reduced the area of summer habitat for monarchs. They need milkweeds for the larvae to eat and wild flowering plants to provide nectar for the adults. There are currently 30 million more acres of corn and soybeans grown in the United States than in 1996. Of this, over 24 million acres are former CRP, grassland, rangeland and wetland habitats that once supported milkweed, monarchs and other wildlife. Weeds in row crops used to be managed with crop rotations and tillage, leaving relatively small quantities of common milkweed that provided good habitat for monarch reproduction. With the adoption of herbicide-tolerant crops, widespread application of glyphosate has all but eliminated milkweeds from row crop areas. The increase in row crop acreage in the U.S. is largely due to the ethanol mandate and subsidies passed by Congress in the 2007 Clean Energy Act. Corn acreage has increased every year since then except for a slight decline in the last two years when more soybeans were planted. About 40 percent of corn grown in the U.S. now goes to make ethanol. Farmers have been plowing up highly erodible land, removing hedgerows, narrowing field margins and planting through grassed waterways. In addition to eliminating milkweed habitat for monarch butterflies, plowing up marginal land to grow corn and soybeans has led to massive erosion of topsoil readily apparent around here this wet spring. I consider this mining, not agriculture. It’s greedy short-sighted taking of soil fertility from future generations that is also ruining wildlife habitat and spoiling our rivers and lakes. Monarch butterfly populations can rebound if they get some good weather during their migrations, if their over-wintering areas in Mexico are protected and if there are enough milkweeds to allow them to reproduce. We can help by protecting grassland areas with established milkweed, by limiting how much we mow, by restoring prairies and by replanting grassed waterways. We can plant milkweeds in wildflower gardens that attract monarchs and other pollinating insects. It just wouldn’t be summer without monarchs winging it across the fields. Please send any comments and suggestions for this column to me at [email protected].

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#### Transcript File - Rights4Bacteria ```Learning Objective:  Understand how to calculate concentration from reacting volumes Learning Outcome:  Use balanced symbol equations to calculate the concentration of an unknown reactant (A/A*) HCl Concentration ? Volume = your average titre NaOH Concentration = 0.1 mol dm-3 Volume = 25 cm3  Learn this formula triangle! number of moles n concentration (in mol dm-3) c v volume (in dm3) 1 litre = 1000 cm3 = 1 dm3  Concentration is a measure of how crowded things are.  The concentration can be measured in moles per dm3 (ie. moles per litre).  So 1 mole of ‘stuff’ in 1dm3 of solution has a concentration of 1 mole per dm3 (1mol/dm3).  The more solute you dissolve in a given volume, the more crowded the solute molecules are and the more concentrated the solution.  Example 1: What is the concentration of a solution with 2 moles of salt in 500cm3? The question already tells us the number of moles and the volume, so use the formula: c = n = 2 = 4 mol/dm3 convert the v 0.5 volume to dm3 first by dividing by 1000.  Example 2: How many moles of sodium chloride are in 250cm3 of a 3 mol dm-3 solution of sodium chloride? The question tells us the volume and concentration, so use the formula: n = c x v = 3 x 0.25 = 0.75 moles convert the volume to dm3 first by dividing by 1000. “In a titration, 20 cm3 of 1.0 mol dm-3 hydrochloric acid, HCl, reacted with 25 cm3 of sodium hydroxide, NaOH. What was the concentration of the sodium hydroxide?”  You will need to write a balanced symbol equation.  Use the crib sheets available (2 options)  Hydrochloric acid = HCl  Nitric acid = HNO3  Sulphuric acid = H2SO4  Sodium hydroxide – NaOH  Sodium sulphate = Na2SO4  Sodium nitrate = NaNO3  Phosphoric acid = H3PO4  Sodium phosphate = Na3PO4  Potassium sulphate = K2SO4 Section A 1. 2. 3. 4. 1.6 mol dm-3 0.08 mol dm-3 0.12 mol dm-3 0.912 mol dm-3 Section B 1. 2. 3. 4. 1.6 mol 0.8 mol 2.4 mol 1.0 mol dm-3 dm-3 dm-3 dm-3  Using your average titre from the previous lesson and calculate the concentration of HCl which was used to neutralise 25cm3 of 0.1mol dm-3 NaOH.  Compare your result to the actual value.  Comment on the accuracy of your result. Learning Objective:  Understand how to calculate concentration from reacting volumes Learning Outcome:  Use balanced symbol equations to calculate the concentration of an unknown reactant (A/A*) ```

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Turns out the famously tough and long-lived Bristlecone Pine is is generally unsusceptible to mountain pine beetle infestation. A new study of the ancient trees, including some specimens high on Mount Charleston, shows they are highly resistant to damage from mountain pine beetles, which have killed millions of other types of pine trees across the West in the past decade. Rising temperatures associated with climate change have fueled a beetle explosion throughout the region, including major outbreaks in high-elevation forests, experts say. But in the same warming woodlands where the bugs are killing large numbers of other pine trees, bristlecones have escaped almost completely unscathed, according to findings by the U.S. Forest Service’s Rocky Mountain Research Station and Utah State University, both in Logan, Utah. “We found no bristlecones that had been attacked,” said Barbara Bentz, the Forest Service’s lead scientist on the study. The reason: The iconic old trees are made from high density wood and contain much higher levels of a chemical resin that repels insect invaders — four times the levels found in foxtail pine and eight times the levels found in limber pine, a common host plant for beetles. “We were very surprised to find such incredible defenses in Great Basin bristlecone pine,” Bentz said. “Extreme longevity and past evolutionary experiences have helped this species survive current pressures in a changing climate.” entire article here: Bristlecone pines untroubled by tree-killing beetle invasion, study shows

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We Like to Read with Enthusiasm Rationale: As children develop into independent readers, it is very important for them to read with fluency. A good indicator of reading fluency is a student's ability to tell a story with expression and enthusiasm. In this lesson, students will practice reading with expression so that they will become more creative and eager readers. 1. Shannon, David. Duck on a Bike. Scholastic Inc. 2002. (Have enough for half of the class.) 2. Shannon, David. David Goes to School. Scholastic Inc. 1999. (Have enough for the other half of the students.) Copies of a partner reading expression evaluation sheet with the a. Did your partner change their voice to make the story more interesting? b. What happened in the story? c. What could your partner do to make their story reading better? 1. Introduce the lesson by explaining to the students that when reading stories, it is important to read them with feeling and enthusiasm. Today, we are all going to practice reading with expression 2. Now, I am going to read one passage in two different ways and I want you to tell me which way you think sounds better. The teacher will read a passage from The Grouchy Ladybug in a very unexciting voice: "Good Morning," said the friendly ladybug. "Go away!" shouted the grouchy ladybug. "I want those aphids." "We can share them," suggested the friendly ladybug. "No. They're mine, all mine," screamed the grouchy ladybug. "Or do you want to fight me for them?" Now, I am going to read the passage in a different voice. . The teacher will reread the passage in a very excited and enthusiastic voice. Was the story more interesting when I told it the first or second time? Could you tell a difference in my expressions? Have them explain or give examples of what made it more interesting the second time the passage was read. 3. When you tell a story to someone, the person listening to you gets interested when you change the tone of your voice to imitate the actions that are happening in the story. For example, when something scary is occurring, let your voice get quiet so that the person can feel the anticipation, and when something thrilling is happening let your voice get louder and more exciting to reflect the joy of the passage. The teacher will distribute the books to the students so that half the has one book, and the other half has a different book. Give students a book talk about each of the two books. Duck On a Bike, Have you ever heard of a duck riding a bike??? A duck finds a bicycle to ride. He is showing off his neat tricks when some kids come through on their bikes. The kids do not see duck on his bike. Half of you will read to find out what happens to Duck!!! The other book is about a student who always gets in trouble at school! Do you know any kids like this? The other half of you will read to see if the boy in the book does any better in school. Now, we are going to read our silently. I want you to read the text two times. The first want you to read the text for fun. The second time you read the want you to think about how you would read the text to someone else in a way that would be interesting to him or her. 5. Now, we are going to practice reading with expression. Now I am going to partner you up with someone who had the opposite book that you had. I want each of you to take turns reading your books to each other. When you read, be sure to read with expression to reflect the words in the book, and when you listen to your partner read, I want you to fill out an evaluation sheet of their reading expression.. 6. The teacher will ask the students to write in their journals about their partner's story. Their journal entry should include the main characters, the setting, the problem, and the solution. They should also include how they think reading with expression makes a story more interesting. 7. While the students are writing in their journals, the teacher(with a checklist) will have each student come up and read their favorite passage in the book with expression, and assess their reading. The journal entries will be an assessment of the students' comprehension of their partner's reading ability. Eldredge, J.L. (1995). Teaching Decoding in Holistic Classrooms. Click here to return to Beginnings

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The fact that a global water crisis is underway is likely unfathomable to many people living in developed countries, where getting clean water is as easy as turning on a faucet. But for those living in developing countries, water insecurity is a very real threat. Although more than 70 percent of the earth’s surface is covered by water, only 2.5 percent of it is fresh and less than 1 percent is actually accessible, with the majority of the earth’s fresh water frozen in glaciers and icebergs. In other words, only a minuscule amount of fresh water is available to support a planet of almost seven billion people. Nearly 800 million people in the world live without access to clean water, and 2.5 billion live without adequate sanitation. This means that the world is far from meeting the Millennium Development Goal (MDG) to halve the number of people living without access to improved sanitation facilities by the year 2015. In fact, a recent update on MDG-progress found that the world will likely miss the 2015 goal by about 500 million people. Today, on World Toilet Day, 15 percent of the global population still practices open defecation and nearly 3.5 million people die each year from water and sanitation-related diseases, with 99 percent of those deaths occurring in developing countries. Water-related illnesses kill 4,100 children under the age of five every day, and one child every twenty-one seconds. In June 2013, a Water and Sanitation Program study sponsored by the World Bank found that children living in villages that received sanitation treatment were taller and therefore healthier than children that lived in villages without sanitation treatment. Drawing from their own research and other studies, the authors found that the taller children who benefited from the program were more likely to lead healthy and economically productive lives, and had greater potential to excel in school and work. In India, one of the world’s most populous countries, clean water and sanitation facilities remain in short supply. Of the one billion people in the world that do not regularly use toilets or latrines, nearly 60 percent reside in India. India’s government and international organizations have taken an active role in trying to address this health risk. In 1999, the Indian government launched the Total Sanitation Campaign (TSC), since renamed Nirmal Bharat Abhiyan, to promote better sanitary habits and latrine construction. Water.org, a nonprofit organization, is also at the forefront of addressing the global water crisis. Water.org recently announced an expanded partnership with the Caterpillar Foundation, which will allow the organization to bring its innovative WaterCredit model to three new countries: Indonesia, the Philippines, and Peru. WaterCredit, launched in 2003 in India, involves giving loans to families to help them pay for installing toilets or other forms of water infrastructure in their homes. Women and girls are a “critical component” of the WaterCredit program, comprising 93 percent of its borrowers. This is unsurprising given that women and girls are responsible for 76 percent of water collection worldwide and bear the brunt of the global water crisis. Without easily accessible water, basic tasks typically assigned to women, such as cooking and washing, become daunting challenges that require hours of labor – time that could otherwise be spent earning an income or going to school. Water.org estimates that, worldwide, collecting water costs women 440 million school days a year and 220 million hours each day. The WaterCredit program gives women and girls the dignity of safe and private hygiene, while also freeing up time for education and work. During a panel at the launch of the Water.org-Caterpillar Foundation partnership, actor, philanthropist, and Water.org co-founder Matt Damon explained that Water.org invests in women “because it works. Those are the investments that you get the best return on.” Caterpillar Foundation President Michele Sullivan agreed, stating: “Without question, [investing in women] is the best return on investment for the long term and it helps the country’s GDP…It helps everything in their country when you invest in women and girls.” Programs like WaterCredit empower the world’s poorest, including women and children, to take control of their futures. For every dollar invested in water and sanitation programs, there is an $8 return in time, productivity, and reduced healthcare costs for the average family. Giving a family access to water, one of the most basic necessities, provides them with opportunities that would otherwise be lost.

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Decimal fractions Any fraction can be written in decimal form. Some are easy to remember: 1/10 = 0.1      1/2 = 0.5 1/4 = 0.25      1/5 = 0.2 but you can work out any decimal fraction by dividing the numerator by the denominator. e.g. 1/8 = 1.000 ÷ 8 = 0.125 3/4 = 3.00 ÷ 4 = 0.75 Can you work out the decimal fractions for some other fractions? What is special about 1/3 and 1/6 ? Are there any other decimal fractions like this? Percentages A percentage is a fraction represented at a part of 100. Some are easy to remember: e.g. 1/100 = 1%   20/100= 20% 1/2 =50%   1/4 = 25% You can work out any percentage by multiplying a decimal fraction by 100 e.g. 1/10= 0.1 = 10% 1/8 = 0.125 = 12.5% 3/4 = 0.75 = 75% You can use the calculator to find a percentage of a number. e.g. What is 25% (1/4) of 64? 6 4 x 2 5 %

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# Time and speed Welcome to aptitude tricks Problems involving Time, Distance and Speed are solved based on one simple formula. Distance = Speed * Time Which implies → Speed = Distance / Time   and Time = Distance / Speed Let us take a look at some simple examples of distance, time and speed problems. Example 1. A boy walks at a speed of 4 kmph. How much time does he take to walk a distance of 20 km? Solution Time = Distance / speed = 20/4 = 5 hours. Example 2. A cyclist covers a distance of 15 miles in 2 hours. Calculate his speed. Solution Speed = Distance/time = 15/2 = 7.5 miles per hour. Example 3. A car takes 4 hours to cover a distance, if it travels at a speed of 40 mph. What should be its speed to cover the same distance in 1.5 hours? Solution Distance covered = 4*40 = 160 miles Speed required to cover the same distance in 1.5 hours = 160/1.5 = 106.66 mph Now, take a look at the following example: Example 4. If a person walks at 4 mph, he covers a certain distance. If he walks at 9 mph, he covers 7.5 miles more. How much distance did he actually cover? Now we can see that the direct application of our usual formula Distance = Speed * Time or its variations cannot be done in this case and we need to put in extra effort to calculate the given parameters. Let us see how this question can be solved. Solution For these kinds of questions, a table like this might make it easier to solve. Distance Speed Time d 4 t d+7.5 9 t Let the distance covered by that person be ‘d’. Walking at 4 mph and covering a distance ‘d’ is done in a time of ‘d/4’ IF he walks at 9 mph, he covers 7.5 miles more than the actual distance d, which is ‘d+7.5’. He does this in a time of (d+7.5)/9. Since the time is same in both the cases → d/4 = (d+7.5)/9            →        9d = 4(d+7.5)   →        9d=4d+30        →        d = 6. So, he covered a distance of 6 miles in 1.5 hours. Example 5. A train is going at 1/3 of its usual speed and it takes an extra 30 minutes to reach its destination. Find its usual time to cover the same distance. Solution Here, we see that the distance is same. Let us assume that its usual speed is ‘s’ and time is ‘t’, then Distance Speed Time d s t min d S+1/3 t+30 min s*t = (1/3)s*(t+30)      →        t = t/3 + 10      →        t = 15. So the actual time taken to cover the distance is 15 minutes. Note: Note the time is expressed in terms of ‘minutes’. When we express distance in terms of miles or kilometers, time is expressed in terms of hours and has to be converted into appropriate units of measurement. Previous Post Next Post

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# Question #28264 Mar 29, 2017 #### Explanation: Here's how to derive the law from Boyle's Law and Charles' Law. Consider an ideal gas at conditions ${p}_{1} , {V}_{1} , {T}_{1}$. Now, keep $T$ constant and vary $p$ and $V$ to bring the gas to a second state ${p}_{2} , V , {T}_{1}$. According to Boyle's Law: (1) ${p}_{1} {V}_{1} = {p}_{2} V$ Now, keep $p$ constant and vary $V$ and $T$ to bring the gas to a third state ${p}_{1} , {V}_{2} , {T}_{2}$. According to Charles Law, (2) $\frac{V}{T} _ 1 = {V}_{2} / {T}_{2}$ From (1), (3) $V = \frac{{p}_{1} {V}_{1}}{p} _ 2$ From (2) (4) $V = \frac{{V}_{2} {T}_{1}}{T} _ 2$ Equating the right hand sides of (3) and (4), we get $\frac{{p}_{1} {V}_{1}}{p} _ 2 = \frac{{V}_{2} {T}_{1}}{T} _ 2$ or $\frac{{p}_{1} {V}_{1}}{T} _ 1 = \frac{{p}_{2} {V}_{2}}{T} _ 2 = {k}^{'}$ (a constant) In general, we can write this as $\frac{p V}{T} = k '$ or $p = \left({k}^{'} / V\right) T$ Now, if we hold the volume $V$ constant, and let ${k}^{'} / V = k$, we get $p = k T$, which is Gay-Lussac's Law.

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A songbird family tree reveals that the earliest songbird was probably a vocal diva Spring will be here soon. And with daffodils, crocuses and other signs of spring comes a burst of birdsong as males duke it out to get female attention. While the males trill loud songs, the females sit quietly, choosing who will be the lucky male. Vocal male and quiet female songbirds are common in temperate zones, and have given rise to a common assumption. The best male songs get picked for reproduction, and this sexual selection results in complex song; females just listen and choose, so female song should be rare. After all, females don’t need to sing to attract mates. But it turns out this commonly held assumption is not true. A new study shows that the majority of females of songbird species do sing, and it’s likely that the ancestor of modern songbirds was also a vocal diva. The results challenge the old wisdom about female songbirds, and suggest that when it comes to female song, it’s not all about sex. Karan Odom, a behavioral ecologist at the University of Maryland, Baltimore County, has always been interested in birdsong. “As I began to study it in depth,” she says, “I realized there was a lot that’s unknown, and one area was the extent to which females were singing and the role that song plays in males and females.” Odom and her colleagues did a survey of 44 songbird families, going through bird handbooks and other sources to find records of whether males, females or both were singers. In results published March 4 in Nature Communications, they showed that female melodies are not rare at all. In fact, 71 percent of the species surveyed have singing ladies. So much for that quiet, retiring female bird. The scientists then mapped the bird species on a phylogenetic tree, a family tree of sorts for a particular group of organisms. By putting species, or family members, in their correct places on the family tree, you can divine what their ancestors may have been like, even if you have never seen that ancestor, notes Mike Webster, an ornithologist at Cornell University in Ithaca, N.Y. “If you had a whole bunch of relatives,” he says, “some with blonde hair and some with brown hair, you map that on the family tree, and you can see that blonde hair originates with one particular person.” We may not know what great-uncle Moe looked like, but if all of his descendants had brown hair, there’s a high likelihood that he did, too. Modern phylogenetic trees are often based on DNA from species that have been sequenced. Scientists can examine areas in the DNA where one letter might have been replaced with another. The similarities and differences between the letters can help determine how closely species are related. The scientists took a phylogenetic tree of songbirdsand looked at each species, noting which had been observed to have female song and which hadn’t. By putting all of the song records on the tree to observe how closely related various species were, Odom and colleagues were able to show that the ancestor species of all songbirds probably had female singers. Kevin Omland, an evolutionary biologist at the University of Maryland and a coauthor on the paper, says that it is far more likely that a small and closely related songbird group lost female song than it is that 71 percent of all songbird species to gained female song through selection. The scientists say that idea that female songbirds don’t sing probably arose not from the songbirds themselves, but from which species were studied and where. “When people first began studying birdsong,” Odom explains, “a lot of them were in temperate regions where not as many female birds sing.” This gave rise to the assumption that singing females were rare. Marlene Zuk, a behavioral ecologist at the University of Minnesota, Twin Cities, is “always a fan of studies that examine commonly held assumptions.” She hopes the results of this paper will cause scientists to examine their ideas a little more. “Not everything is the same as in the temperate zone,” she notes. “There could be other implicit assumptions we are not aware of.” Now that Odom knows more female songbirds sing, she wants to dig deeper to find out why. Females don’t have to sing to attract mates, but Odom hypothesizes that “maybe some other selection pressures are at play: to defend a territory or compete for resources.” With the new understanding of how widespread female song is, that’s a question that doesn’t have to be left for the birds.

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This post assumes familiarity with “Two’s Complement”, “One’s Complement”  and an understanding of “Positional Numeral Systems”. Why is the Two’s Complement called the Two’s Complement? Ever wondered why the two’s complement and the one’s complement are named as such? We were told that to calculate the two’s complement of a number, you add 1 to its one’s complement. But why? When calculating the one’s complement, we simply subtract every digit from 1, so why don’t we subtract every digit from 2 in two’s complement? And maybe, you tried to reason with yourself about how there’s no “2” in the binary system and so that mehtod would not work anyway. There is however a much better way to understand and explain it. To understand this, you need to understand the difference between the radix complement and the diminished radix complement. According to wiktionary: • The radix complement is the number which, when added to an n-digit number in radix-r, results in r^n. An alternative way of looking at it is that it is the smallest possible (n-1)-digit number in radix-r. The radix complement for radix-r is called r’s complement. We get it by adding 1 to the diminished radix complement. • The diminished radix complement is the number which, when added to an n-digit number in radix-r results in r^n -1. An alternative way of looking at is is that it is the largest possible n-digit number in radix-r. The diminished radix complement for radix-r is called (r-1)’s complement. We get it by subtracting every digit from (r-1) In radix-2, the radix complement is the two’s complement and the diminished radix complement is the one’s complement. Let’s try it for an arbitrary byte. The one’s complement of (11001011) is (00110100). If we add them together, we get (11111111) which is the largest 8-digit binary number. The two’s complement however is (00110101). If we add it to (11001011), we get 100000000 which is the smallest 9-digit binary number. Let’s try the same with the popular and intuitive decimal system. Let’s take 589 as our arbitrary decimal number. The diminished radix complement of 589 aka the nine’s complement would be 410. If we add 589 and 410, we get 999 which is the largest possible 3-digit decimal number. The radix complement aka the ten’s complement would be 411. If we add together 589 and 411 we get 1000 which is the smallest possible 4-digit number. Just to clear concepts, let’s try ternary(or trinary aka radix-3) as well. The radix complement would be the three’s complement whereas the diminished radix complement would be the two’s complement. Our number is 1021. It’s two’s complement will be 1201. Adding together 1021 and 1201 we get 2222 which is the largest possible 4-digit number in radix-3. It’s three’s complement will be 1202 which, when added to 1021 gives 10000 which is the smallest possible 5-digit number in radix-3. You must have noticed how the method for calculating the two’s complement in radix-3 was different from the method for calculating the two’s complement in radix-2. Because in binary, the two’s complement is the radix complement whereas in ternary, it is the diminished radix complement. The two share the same name but are completely independent of one another. Two’s Complement vs Two’s Complement Notation People, especially CS undergraduates, confuse the two when they are actually two distinct concepts. A junior at my university once told me that she once got a question in an exam in which she was supposed to write the two’s complement of 27 and according to her, it was a trick question because 27 cannot possibly have a two’s complement as two’s complements exist only for negative numbers. I was like what? Every binary number has a two’s complement. The two’s complement of 1011 is 0101 and the two’s complement of 0101 is 1011. It has nothing to do with negative numbers. Then what is the deal with negatives having two’s complements? Well, computers have different ways of storing and identifying negative numbers and one of them is called two’s complement notation. In two’s complement notation, for n digits, all numbers between 0 and 10^(n-1) represent positive numbers and all numbers from 10^(n-1) to 10^n -1 represent negative numbers. So all binary numbers that start with a 1 represent negative values of their twos complement. in 4 digits, 0111 represents 7 while 1001, its two’s complement, represents -7. In binary, 27 is 011011, whose two’s complement is 100101. In two’s complement notation, 27 is still 11011, however -27 will be 100101.

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Potassium argon dating advantages The isotopes the KAr system relies on are Potassium (K) and Argon (Ar).Potassium, an alkali metal, the Earth's eighth most abundant element is common in many rocks and rock-forming minerals.Dating of movement on fault systems is also possible with the Ar method. What simplifies things is that potassium is a reactive metal and argon is an inert gas: Potassium is always tightly locked up in minerals whereas argon is not part of any minerals. So assuming that no air gets into a mineral grain when it first forms, it has zero argon content.Potassium (K) is one of the most abundant elements in the Earth's crust (2.4% by mass).One out of every 10,000 Potassium atoms is radioactive Potassium-40 (K-40).Heating causes the crystal structure of the mineral (or minerals) to degrade, and, as the sample melts, trapped gases are released. The gas may include atmospheric gases, such as carbon dioxide, water, nitrogen, and argon, and radiogenic gases, like argon and helium, generated from regular radioactive decay over geologic time.Thus, a granite containing all three minerals will record three different "ages" of emplacement as it cools down through these closure temperatures.

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Often avoiding sampling gear with their capability to detect movements and swim their way out of the nets fast enough, the small squids living in the open-ocean zone have so long gone under-researched. The present study, conducted by Dr. Michel Hendrickx, Universidad Nacional Autonoma de Mexico, and his team, seems to provide new and first distributional records of five such species for the Gulf of California and in southwestern Mexico. It also significantly expands the currently known southernmost limit of localities of some of these squids in the eastern Pacific. The research is available in the open-access journal ZooKeys. The researched five squid species belong to two genera, Abraliopsis and Pterygioteuthis, which although abundant and diverse, have long been shrouded by taxonomic and distributional controversies. To solve them, the researchers used specimens, collected over the span of thirteen years, comprising eight cruises across 113 locations in the Gulf of California and off the southwestern coast of Mexico. As a result, the scientists concluded a significantly wider distributional range of the species they found. For instance, squids of the Abraliopsis genus were surprisingly found in water deeper than 600 m during the day. The studied small squids are of high ecological value due to their vital position in the food web. Members of Abraliopsis are important preys for many fishes and mammals, such as the peruvian hake, the Indo-Pacific sailfish, the common dolphinfish and the local sharks. Meanwhile, the representatives of the other researched genus, Pterygioteuthis, are often consumed by larger cephalopods, sea-birds and fur seals. However, their abundance is strongly dependent on temperature, especially when there are fast and significant changes. The scientists suggest that additional samplings with more adequate equipment, like faster large-sized mid-water trawls, could further bridge the knowledge gaps about these elusive marine inhabitants. Hendrickx ME, Urbano B, Zamorano P (2015) Distribution of pelagic squids Abraliopsis Joubin, 1896 (Enoploteuthidae) and Pterygioteuthis P. Fischer, 1896 (Pyroteuthidae) (Cephalopoda, Decapodiformes, Oegopsida) in the Mexican Pacific. ZooKeys 537: 51-64. doi: 10.3897/zookeys.537.6023.

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Photo by Husniati Salma on Unsplash Compassion and education for everyone are closely related in several ways. Here are some of the connections between the two: - Inclusive learning environments: Education that embraces compassion creates inclusive learning environments where all individuals, regardless of their background, abilities, or circumstances, feel valued and respected. Compassion promotes empathy, understanding, and acceptance, fostering an atmosphere that encourages collaboration, diversity, and equal opportunities for everyone to learn and thrive. - Emotional well-being: Compassion in education recognizes the importance of addressing students' emotional well-being. When educators and educational institutions prioritize compassion, they create a supportive and nurturing environment that acknowledges and responds to students' emotional needs. Compassionate educators provide a safe space for students to express their feelings, develop emotional intelligence, and build resilience. - Empathy and understanding: Education that cultivates compassion helps students develop empathy and understanding towards others. By teaching about diverse perspectives, cultures, and experiences, education promotes empathy and helps students appreciate the common humanity we all share. Compassionate education encourages students to put themselves in others' shoes, fostering kindness, respect, and a sense of interconnectedness. - Building positive relationships: Compassion strengthens the relationships between students, teachers, and the entire school community. When educators model and promote compassionate behavior, they encourage positive interactions, cooperation, and a sense of belonging among students. Compassionate education emphasizes the importance of treating others with kindness and respect, fostering healthy and supportive relationships within the educational setting. - Conflict resolution and empathy skills: Compassion is crucial for teaching conflict resolution and empathy skills. By integrating compassion into educational practices, educators can help students develop the ability to resolve conflicts peacefully, communicate effectively, and understand differing viewpoints. Compassionate education encourages dialogue, active listening, and finding solutions that prioritize the well-being of all individuals involved. - Social justice and equity: Compassion is closely tied to promoting social justice and equity in education. A compassionate education system recognizes and addresses systemic inequalities, ensuring that all students have access to quality education and equal opportunities to succeed. It involves dismantling barriers, challenging biases, and fostering a sense of social responsibility among students to create a more just and inclusive society. - Cultivating responsible citizens: Education that incorporates compassion aims to cultivate responsible and compassionate citizens. By teaching values such as empathy, kindness, and respect, education prepares students to contribute positively to their communities and make a difference in the world. Compassionate education encourages active citizenship, social engagement, and a commitment to creating a more compassionate and just society. In summary, compassion and education for everyone are intertwined concepts. Compassionate education creates inclusive, nurturing, and empathetic learning environments, fosters positive relationships, promotes social justice, and prepares students to become responsible and compassionate individuals. By integrating compassion into education, we can empower individuals to make a positive impact on their own lives and the lives of others.

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By Emma Johnston Seventy-five percent of all arable land is used to support animals other than humans. However, it takes less land to grow crops that humans can directly eat than it does to raise cattle, sheep and other animals, researchers say. Prof. Christine Costello, University of Missouri, bioengineering, a former postdoctoral researcher in ecology and evolutionary biology at Cornell, found that the United States could both trim its population’s waistlines and reduce its environmental impact by swapping some of its meat consumption for consumption of vegetables, milk and eggs. According to Emily Cassidy grad, University of Minnesota, growing crops instead of raising animals could feed as many as four billion more people. Costello compared greenhouse gas emissions, land use and use of nitrogen in soil associated with different diet types. These diets include the average American diet, the omnivore, diet recommended by the United States Department of Agriculture, a Demitarian diet, which reduces average meat consumption by half, a ovo-lacto vegetarian diet that includes eggs and dairy and a vegan diet. Costello used a life-cycle analysis to measure the environmental impact, land use and nitrogen and fertilizer input necessary for each diet at all stages of production. According to Costello, it takes 4,150 square meters of land to produce all the food for an average American adult for a year, 3,370 square meters for the recommended omnivore and 890 square meters for the ovo-lacto vegetarian. This means that with the same amount of land required to feed the average American, almost five ovo-lacto vegetarians could be fed. Increasing trade, however, complicates the conversation of food production, according to Costello. If a farm in the Midwest exports goods to the coastal U.S., is it fair for the Gulf States to incur the bulk of environmental damages? With problems like nitrogen pollution, should environmental harm be attributed to the exporter or the consumer? Costello said that nitrogen pollution does not seem to be an issue because nitrogen is an invisible gas. “Nobody’s seeing the environmental impacts; it’s a problem for another region,” Costello said. Nitrogen use in farming is a double-edged sword, it both improves yields because it is a necessary soil nutrient, but too much can cause environmental problems such as algal blooms. “The green revolution has done amazing, great things for humanity, but now we’re dealing with the dead zone in the Gulf of Mexico or groundwater with really high nitrate levels,” Costello said. According to Costello, the average U.S. diet requires twice as much nitrogen as a vegetarian dietiet. As a leading food producer for the world, the United States has decreased its beef consumption in recent years, but production of beef has not followed this trend because the United States exports what it does not eat to other countries. According to the USDA, the average person in the U.S. still eats twice as much protein than is recommended, and more than half of that protein comes from meat. According to Cassidy, this helps explain why it takes 16.1 hectares to feed a person in the U.S. but only 6.5 hectares to feed a person in India. According to Costello, food waste in the U.S. is also a problem. Close to half of all fruits, vegetables, and dairy products, 35 percent of poultry and 20 percent of beef are wasted at the consumer level, according to the USDA. Costello advocates reducing food waste by suggesting that individuals investigate what is being grown in the first place and what implications thier food choices have for nitrogen pollution and improving health. “Sustainability is about being where we’re maximizing human well-being, where we’re not destroying the ecosystems that we need for food and not constantly facing nutrient-related illnesses, whether malnutrition or over-nutrition,” Costello said. “I wouldn’t want to take away from the ability of a child who’s malnourished to eat something that’s really nutritious like milk or beef. That’s not sustainable.”

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## Introduction: ☠WEEDINATOR☠ Part 4: Differential Steering Geometry Code If you have the time to watch the above video, you'll notice that there are some strange noises caused by the motors on the steering stalling every now and again as the WEEDINATOR navigates a 3 point turn. The motors are essentially jamming against each other as the radius of turn is different on the inside to the outside and the distance the wheel travels is different per degree of turn. The geometry of the turn can be worked out by sketching out 8 or so permutations of the turn, giving examples of turning at different angles on the inner wheel from 0 (no turn) to 90 (full lock) degrees. Sounds complicated? Most small wheeled robots don't attempt to have any kind of sophisticated steering and rely, very effectively, on simply changing the relative speed of the motors on each side of the vehicle, which is pretty much the same as how a tracked digger or tank works. This is great if you're charging over a crater filled war zone shooting at everything that moves, but in a tranquil agricultural environment it's important to do as little damage to the soil and ground as possible so grinding wheels back and forwards against each other is not appropriate! Most cars and tractors have a very useful gadget called a 'Differential', except the cars you see in old American movies where you can hear the tyres screeching like crazy every time they go round a corner. Do Americans still build cars like this? With the WEEDINATOR, we can program differential into the drive motors by working out the formula for the relative speeds and angles of the wheels at any particular angle of turn. Still sounds complicated? Here's a quick example: If the WEEDINATOR is navigating a turn and has it's inside wheel at 45 degrees, the outside wheel is NOT 45 degrees, it's more like 30 degrees. Also, the inside wheel may be turning at 1 km/hour, but the outside wheel will be significantly faster, more like 1.35 km/hour. ## Step 1: Geometry Setup A few basic assumptions are made to begin with: • The chassis will pivot about one of the back wheels as shown in the diagram above. • The effective centre of the pivot circle will move along a line extended from the centres of the two back wheels, depending on the angle of turn. • The geometry will take the form of a sine curve. ## Step 2: Scaled Drawings of Wheel Angles and Radii A full scale drawing was made of the WEEDINATOR front wheels and chassis with 8 different permutations of inside wheel angle between 0 and 90 degrees and the respective turn centres were mapped out as shown in the drawings above. The effective radii were measured from the drawing and plotted on a graph in Microsoft Excel. Two graphs were produced, one of the ratio of the left and right front wheel axles and another for the ratio of the two radii for each particular turn angle. I then 'fudged' up some formulae to mimic the empirical results based on a sine curve. One of the fudgings looks like this: `speedRatio= (sin(inner*1.65*pi/180)+2.7)/2.7; // inner is the inner turn angle.` The curves were fudged by changing the values shown in red in the excel file until the curves fitted together. ## Step 3: Coding the Formulae Rather than trying to code the formulae in one line, they were broken down into 3 stages to allow the Arduino to process the math properly. The results are shown in the serial port display and checked with the measured results on the scale drawing. ### Attachments Participated in the Remote Control Contest 2017 Participated in the Arduino Contest 2017 Participated in the Wheels Contest 2017

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# Implicit Differentiation ## Homework Statement 8x^2-10xy+3y^2=26 2. The attempt at a solution (8)(2x)-(-10x)y'+(y)(-10)+(3)(2y)y'=0 16x+10x(y')-10y+6y(y')=0 y'(10x+6y)+16x-10y=0 y'(10x+6y)=10y-16x y'=(10y-16x)/(10x+6y) y'=(5y-8x)/(5x+3y) I know I'm doing something wrong but I can't see it for myself. Can someome point me into the right direction? rock.freak667 Homework Helper (8)(2x)-(-10x)y'+(y)(-10)+(3)(2y)y'=0 Where'd you get that extra minus sign from when differentiating -10xy? I differentiated (-10xy) rock.freak667 Homework Helper (8)(2x)-(-10x)y'+(y)(-10)+(3)(2y)y'=0 That -ve sign should be a + sign. I'm still getting (5y-8x)/(3y-5x) rock.freak667 Homework Helper I'm still getting (5y-8x)/(3y-5x) What answer are you supposed to get? (8x-5y)/(5x-3y) rock.freak667 Homework Helper (8x-5y)/(5x-3y) Well multiply both the numerator and denominator by -1. Only thing that I see happened was that y' was moved to the right side of the equation therefore changing the signs. After I done it that way, I got the correct answer. Go figure! HallsofIvy Homework Helper (5y-8x)/(3y-5x)= (8x-5y)/(5x-3y) (5y-8x)/(3y-5x)= (8x-5y)/(5x-3y) I have another problem as well. x + (sqrtx)(sqrty) = 2y 1 + (x^1/2)/(2y^1/2)y' + (y^1/2)/(2x^1/2) = 2y' My question is why I suppose to multiply both sides by (2x^1/2 * y^1/2) and not both of the denominators? rock.freak667 Homework Helper I have another problem as well. x + (sqrtx)(sqrty) = 2y 1 + (x^1/2)/(2y^1/2)y' + (y^1/2)/(2x^1/2) = 2y' My question is why I suppose to multiply both sides by (2x^1/2 * y^1/2) and not both of the denominators? the common denominator of those the terms (not the 1 and not the 2y') is 2x1/2y1/2

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A can do a piece of work in 14 days while B can do it in 21 days. Question: A can do a piece of work in 14 days while B can do it in 21 days. They began together and worked at it for 6 days. Then, A fell ill and B had to complete the remaining work alone. In how many days was the work completed? Solution: Time taken by $\mathrm{A}$ to complete the work $=14$ days Work done by $\mathrm{A}$ in one day $=\frac{1}{14}$ Time taken by $\mathrm{B}$ to complete the work $=21$ days Work done by $\mathrm{B}$ in one day $=\frac{1}{21}$ Work done jointly by $\mathrm{A}$ and $\mathrm{B}$ in one day $=\frac{1}{14}+\frac{1}{21}=\frac{3+2}{42}=\frac{5}{42}$ Work done by $\mathrm{A}$ and $\mathrm{B}$ in 6 days $=\frac{5}{42} \times 6=\frac{5}{7}$ Work left $=1-\frac{5}{7}=\frac{2}{7}$ With B working alone, time required to complete the work $=\frac{2}{7} \div \frac{1}{21}=\frac{2}{7} \times 21=2 \times 3=6$ days So, the total time taken to complete the work $=6+6=12$ days

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# Trigonometry math solver One tool that can be used is Trigonometry math solver. We can help me with math work. ## The Best Trigonometry math solver Apps can be a great way to help students with their algebra. Let's try the best Trigonometry math solver. The best geometric sequence solver is a computer program that solves geometric sequences, such as those found in long multiplication problems. The program works by taking a list of numbers and linking them together to produce a longer list. This process is repeated until the sequence is solved. The best geometric sequence solver can work in several ways. It can use either brute force or brute force with some help from a human. It can also use sorting or other computer algorithms to determine the next number in the sequence and find the gap between it and the other numbers. Once all the numbers have been determined, they are combined into one long list, which represents the solution to the problem. There are two main types of geometric sequence solvers. One type uses brute force and tries every possible combination until one of them works. The other type uses brute force with some help from a human and tries every combination that meets certain requirements, such as being in order or not having too many digits. Many people prefer using a geometric sequence solver because it can be faster than using other strategies, such as counting or figuring out how many digits there are in each number in the problem. This makes it great for students who don’t have time to think through their problems carefully or for people who have trouble with math in general. However, some people dislike these programs because they can take longer than typical math problems Logarithmic equations are equations that can be written in the form of a logarithm. For example, if x is the variable and y = log(x), then log(x) = y. This means that the function y = log(x) is a logarithm of the variable x. A logarithm of a variable is a transformation of the variable such that the original value becomes 1, the base 10 value, after being divided by the log base 10 value (base e). Therefore, if x is the variable and y = log(x), then log(x) = y. This means that the function y = log(x) is a logarithm of x. As an example, let's say you're trying to solve an equation like: y = 1000 + 1 + 0.25x You can use a graphing calculator to graph this equation and determine a possible solution is 0.0625 x 0.072125 which means y 0.0625 1000 - 1 + 0.25 1000 - 5 + 0.3125 1000 - 8 + 0.4125 1000 - 975 + 1 and so on... However, using traditional math methods you may get stuck on this problem because you will have to solve for several different values of y, which could There are several ways to solve a problem, but if you’re looking for the best way, then go with the one that has the least amount of steps. It’s always better to have fewer steps than more steps because it saves you time and energy. For example, if you’re trying to get a new computer, then you can just buy one instead of going through an entire process of setting up a computer. It will also save you money because there is no need for you to buy a desk or other furniture. You can also save time by not having to drive from place to place, or sitting in traffic on your way there. There are many other reasons why it’s better to have fewer steps; just think about them and choose the one that fits your situation best. As the name suggests, algebra is a branch of mathematics that deals with mathematical expressions. These expressions may be numerical or symbolic and they usually contain numbers, variables and operators. Further, the most common types of expressions in algebra are polynomials, linear equations, inequalities and rational expressions. A person who studies algebra is known as an algebraist. The best algebrator you can ask for is one that knows what your teacher is looking for. For example, if your teacher asks for a perfect squared sum of c squared plus b squared minus a squared, you could say "57 + 12x - 4y" or "57 + 169x - 243y", but it would be better if the algebrator could recognize this as a perfect squared sum without any extra work on your part; then you could simply enter the answer into your algebrator's calculator. R is a useful tool for solving for radius. Think of it like a ruler. If someone is standing in front of you, you can use your hand to measure their height and then use the same measurement to determine the radius of their arm. For example, if someone is 5 feet tall and has an arm that is 6 inches long, their radius would be 5 inches. The formula for calculating radius looks like this: [ ext{radius} = ext{length} imes ext{9} ] It's really just making the length times 9. So, if they're 6 inches tall and their arm is 6 inches long, their radius would be 36 inches. Using R makes sense when you are trying to solve for any other dimension besides length - such as width or depth. If a chair is 4 feet wide and 3 feet deep, then its width would be equal to half its depth (2 x 3 = 6), so you could easily calculate its width by dividing 2 by 1.5 (6 ÷ 2). But if you were trying to figure out the chair's height instead of its width, you would need an actual ruler to measure the distance between the ground and the seat. The solution to this problem would be easier with R than without it. This is one of the best apps ever I see among of all math solution, the app can't bring more space and they extent their solution with step by step. It’s made our task became easy and we can comfortably do our task without any doubt. Lucille Wilson If you are just tired of solving a math and have no one to make you understand I would definitely recommend to use this app. What you have to do is just click the pic of the problem and they will show you full procedure of solving the problem. This app is super cool. I have used this app for the first time and now I am just loving this after getting understand couple of hard problems just before my math exams. Gretchen Brown Work out math problems Free math website Answer for math Rational inequality solver Get help with math Help me with my math homework

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By Keith Robinson August 23, 2012 Purdue Extension has created a guide to help consumers best protect themselves from the recent outbreak of salmonellosis from cantaloupe. The guide, in a question-and-answer format, explains the Salmonella bacterium that causes foodborne illness, and it offers steps that people can take to help safeguard against it. The Centers for Disease Control and Prevention on Aug. 17 issued an alert about the salmonellosis outbreak linked to two deaths and many illnesses across the country. On Wednesday (Aug. 22), the Food and Drug Administration announced a recall of cantaloupe from Chamberlain Farms in the southwest Indiana community of Owensville in Gibson County because it may be contaminated with Salmonella associated with the outbreak. The CDC continues to work with the FDA and local health officials in the affected states to investigate the outbreak and determine if there are additional sources. "Since the announcement of the outbreak last week, we know consumers, growers and marketers have questions about their particular situation, and it can be hard to find answers," said Extension horticultural specialist Liz Maynard. "We wanted to pull together information together and provide it in a single place. "There are cantaloupes, watermelons and specialty melons grown across Indiana; sold at farm stands, farmers markets, produce auctions and grocery stores; and enjoyed by many as part of a healthy diet. We want to make it easy for individuals and businesses to get the latest information that will help everyone stay healthy and continue to enjoy Indiana melons." Below are excerpts from the Q&A guide, which is available online at https://ag.purdue.edu/hla/fruitveg/Documents/outbreak2012/SalmonellaQA.pdf (PDF: 154 KB). It is based on current information, which will be updated as needed. Question: Where did the Salmonella come from? How did it get on the cantaloupes? Answer: We don't yet know where the particular Salmonella that caused this outbreak came from. However, there are some basic things we do know about how Salmonella can get on produce. Salmonella is common in the environment. A cantaloupe could become contaminated in the field if it came in contact with animal feces or soil, or it could be contaminated during or after harvest through contact with a person, equipment or water that was contaminated with Salmonella. Good agricultural and sanitation practices, such as applying manure fertilizer long before crops are planted, ensuring all employees wash hands, and using clean water for irrigation and for washing produce, can minimize the possibility of contamination. Question: How do I know if a cantaloupe is from Chamberlain Farms? Answer: Ask the retailer or wholesale distributor if the cantaloupe came from Chamberlain Farms. If it did, do not eat the cantaloupe and do not feed it to animals. Put the cantaloupe in a plastic bag and put it in a sealed trashcan so that animals cannot eat it. More information is available from the CDC. Question: I have a cantaloupe that isn't from Chamberlain Farms. What should I do with it? Answer: Cantaloupes from other farms have not been recalled. Follow recommended practices for washing, handling and storing cantaloupe before eating it. As with any fresh produce, cut away any damaged or bruised areas and wash cantaloupe thoroughly under running water before eating or cutting. Washing with soap or detergent or using commercial produce washes is not recommended. Although you will remove the rind before eating, it is still important to wash it first so dirt and bacteria aren't transferred from the knife onto the flesh of the melon. Scrub the rind with a clean produce brush before cutting. Dry the cantaloupe with a clean cloth towel or paper towel to further reduce bacteria that may be present. Question: Is the cantaloupe I bought from my local farmers market safe to eat? Answer: When you purchase cantaloupe from a farmers market, fruit stand or other outlet, ask the vendor where the cantaloupe came from. Unless the cantaloupe was grown on the farm involved in the recall, there is no specific concern.

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# Category: Teaching Graphs ## Enjoy Making Bar Graphs Using graphs to represent data is an important feature of teaching math to elementary school students. Graphs come in many different shapes and sizes and can convey numerous types of information. As students progress through Math, they will encounter graphs in increasing complexity and will be asked to interpret data from graphs, draw conclusions from graphs, and even extrapolate information. Introducing graphing can be done through fun, interactive games that bring Math to life.  (photo: www.commons.wikimedia.org/wiki/File:3D_Bar_Graph_Meeting.jpg (10/27/13 7:15pm) In this activity students will explore the how to collect and graph discrete data. Materials:   Chart Paper, Markers Instructions: 1. In the classroom, section off 4 distinct areas. Using corners of the classroom is the easiest way to do this. 2. Ask the students: “What is your favorite dessert? If you like cake, go to Corner #1; if you like cookies, go to Corner #2; if you like ice cream, go to Corner #3; if you like candy, go to Corner #4. 3. Allow time for students to decide which dessert they prefer and then record the number of students in each corner. 4. On the chart paper, have 4 columns, one for each dessert option. Write the number of students in the corresponding column. 5. Explain to the class that they just collected data on the type of dessert their classmates like. You may consider saying, “Data can be in the form of numbers or words, and in this case, we determined how many of you like each type of dessert. Next, we are going to do a graph, which is similar to a picture, showing the data we just collected.” 6. Create the axis of the graph, labeling the number of students on the vertical axis (y-axis) and the type of dessert on the horizontal axis (x-axis). 7. Mark the y-axis according to provide enough numbers to represent the numbers of students in each category. Draw in the bars to the corresponding number for each dessert type. For younger students, consider using stickers to represent the bars of the graph and have each student place a sticker in the dessert column they prefer. Graphing is essential to building scientific knowledge and understanding as well as Math comprehension. For more interesting graphing activities, visit: And for more of our Fun Learning Math Games, you can visit here: www.math-lessons.ca/activities/index.html www.math-lessons.ca/activities/FractionsBoard5.html www.math-lessons.ca/activities/Geometry.html http://www.literature-enrichment.com/ ## Heart Math What colors of candy are more popular in a typical bag of Valentine Hearts?  World over, kids pretty much enjoy receiving and giving Valentines to their friends on Valentine’s Day.  This year, make it a math learning experience, so the fun is included in the work. Here is a Fun Idea for making it a Happy Heart Math day applicable to Grades 1-4 that encourage comprehension skills of: Assessing, making predictions, and organizing

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This year I set a challenge to make my students aware of the fact that English is packed with Phrasal verbs and it is what keeps them away from sounding more natural or fluent. Most of them have seen Phrasal Verbs before but have not really paid that much attention or went through the 'noticing stage'. How I teach phrasal verbs? - Present them ALWAYS in context, short dialogues or sentences with clear meaning. - Get them to think of the meaning individually and then share their thoughts in pairs. - Get them to look up for more examples or prepare some yourself beforehand. - Analyze the examples and the phrasal verbs. Is it possible to insert a word between a verb and a particle? - Ask your students to create a small dialogue or a situation using the phrasal verb. - Students take turns to act it out. - Next lesson: Students read out their dialogue again without saying the phrasal verb. The rest of the class listens to the dialogue and guesses the phrasal verb. OR get them to act it out using body language and words.

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