{ "log_file": [ "step_121-b_0.log", "step_121-b_1.log", "step_121-b_2.log", "step_121-b_3.log", "step_121-b_4.log", "step_121-b_5.log" ], "query": [ "Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.\n\n### Response:\n", "Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.\n\n### Response:\nSEARCH[\"volume\", \"sphere\", \"formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/698.json\n\n#### Similar Question\nA large sphere has a volume of $288\\pi$ cubic units. A smaller sphere has a volume which is $12.5\\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction.\n\n#### User Answer\nWe know that the two spheres are similar (in the same sense that triangles are similar) because corresponding parts are in proportion. We will prove that for two spheres that are similar in the ratio $1:k$, their volumes have the ratio $1:k^3$. Let the radius of the first sphere be $r$, so the radius of the other sphere is $kr$. The volume of the first sphere is $\\frac{4}{3}\\pi r^3$ and the volume of the second sphere is $\\frac{4}{3}\\pi (kr)^3$. The ratio between the two volumes is \\[\\frac{\\frac{4}{3}\\pi r^3}{\\frac{4}{3}\\pi (kr)^3}=\\frac{r^3}{k^3r^3}=\\frac{1}{k^3}\\] Thus, the ratio of the volumes of the two spheres is $1:k^3$.\n\nIn this problem, since the smaller sphere has $12.5\\%=\\frac{1}{8}$ of the volume of the larger sphere, the radius is $\\sqrt[3]{\\frac{1}{8}}=\\frac{1}{2}$ that of the larger sphere. Thus, the ratio between the two radii is $\\frac{1}{2}$.\n\n(In general, the ratio of the volumes of two similar 3-D shapes is the cube of the ratio of the lengths of corresponding sides.)\n\n--- RESULTS END ---\n\n", "Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.\n\n### Response:\n", "Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.\n\n### Response:\nSEARCH[\"volume\", \"sphere\", \"formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/724.json\n\n#### Similar Question\nA sphere is cut into four congruent wedges. The circumference of the sphere is $12\\pi$ inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of $\\pi$.\n\nNote: To measure the circumference, take the largest circle on the surface of the sphere.\n\n#### User Answer\nLet the sphere's radius be $r$. From the sphere's circumference we have $2\\pi r = 12\\pi$; solving for $r$ yields $r = 6$. The volume of the sphere is $\\frac{4}{3}\\pi r^3 = \\frac{4}{3}\\pi (6^3) = 36\\cdot 8 \\pi$. The volume of one wedge is one-fourth this volume, or $\\frac{1}{4} \\cdot 6^2\\cdot 8 \\pi = 6^2\\cdot 2\\pi = 72\\pi$.\n\n--- RESULTS END ---\n\n", "Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.\n\n### Response:\n", "Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nThe volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.\n\n### Response:\nSEARCH[\"volume\", \"sphere\", \"formula\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/1086.json\n\n#### Similar Question\nThe radius of a sphere is $p$ units and the radius of a hemisphere is $2p$ units. What is the ratio of the volume of the sphere to the volume of the hemisphere?\n\n#### User Answer\nThe volume of the sphere is \\[\\frac{4}{3}\\pi p^3\\] and the volume of the hemisphere is \\[\\frac{1}{2}\\cdot \\frac{4}{3}\\pi (2p)^3 = \\frac{4}{3}\\pi p^3 \\cdot 4.\\] Thus the ratio of the volume of the sphere to the volume of the hemisphere is $\\frac{1}{4}$.\n\n--- RESULTS END ---\n\n" ], "response": [ " SEARCH[\"volume\", \"sphere\", \"formula\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. Hence, $\\frac{4}{3}\\pi r^3 = 36\\pi$ or $r^3 = 9$. Thus, $r = 3$, and the surface area is $4\\pi r^2 = \\boxed{36\\pi}$.", " SEARCH[\"volume\", \"sphere\", \"formula\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe formula for the volume of a sphere is $\\frac{4}{3}\\pi r^3$. Here, we are given that the volume is $36\\pi$, so we have $\\frac{4}{3}\\pi r^3 = 36\\pi$. Solving for $r$ yields $r = 6$. The surface area of a sphere with radius $r$ is $4\\pi r^2$, so the surface area is $4\\pi \\cdot 6^2 = \\boxed{144\\pi}$.", " SEARCH[\"volume\", \"sphere\", \"formula\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. We set this equal to 36 and solve for $r$. This gives us $r^3=\\frac{36}{4}\\pi=9\\pi$, so $r=3$. The surface area of a sphere with radius $r$ is $4\\pi r^2$, so our sphere has surface area $4\\pi(3^2)=\\boxed{36\\pi}$ square inches." ], "src_path": [ "train/geometry/82.json", "train/geometry/82.json", "train/geometry/82.json", "train/geometry/82.json", "train/geometry/82.json", "train/geometry/82.json" ], "input": [ "The volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.", "The volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.", "The volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.", "The volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.", "The volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$.", "The volume of a given sphere is $36\\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\\pi$." ], "output": [ "The volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. Here, we have $\\frac{4}{3}\\pi r^3=36\\pi$. Dividing both sides by $\\frac{4}{3}\\pi$ yields $r^3 = 27$; cube rooting both sides yields $r=3$. The surface area of a sphere with radius $r$ is $4\\pi r^2$; here, our surface area is $4\\pi(3^2)=\\boxed{36\\pi}$.", "The volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. Here, we have $\\frac{4}{3}\\pi r^3=36\\pi$. Dividing both sides by $\\frac{4}{3}\\pi$ yields $r^3 = 27$; cube rooting both sides yields $r=3$. The surface area of a sphere with radius $r$ is $4\\pi r^2$; here, our surface area is $4\\pi(3^2)=\\boxed{36\\pi}$.", "The volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. Here, we have $\\frac{4}{3}\\pi r^3=36\\pi$. Dividing both sides by $\\frac{4}{3}\\pi$ yields $r^3 = 27$; cube rooting both sides yields $r=3$. The surface area of a sphere with radius $r$ is $4\\pi r^2$; here, our surface area is $4\\pi(3^2)=\\boxed{36\\pi}$.", "The volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. Here, we have $\\frac{4}{3}\\pi r^3=36\\pi$. Dividing both sides by $\\frac{4}{3}\\pi$ yields $r^3 = 27$; cube rooting both sides yields $r=3$. The surface area of a sphere with radius $r$ is $4\\pi r^2$; here, our surface area is $4\\pi(3^2)=\\boxed{36\\pi}$.", "The volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. Here, we have $\\frac{4}{3}\\pi r^3=36\\pi$. Dividing both sides by $\\frac{4}{3}\\pi$ yields $r^3 = 27$; cube rooting both sides yields $r=3$. The surface area of a sphere with radius $r$ is $4\\pi r^2$; here, our surface area is $4\\pi(3^2)=\\boxed{36\\pi}$.", "The volume of a sphere with radius $r$ is $\\frac{4}{3}\\pi r^3$. Here, we have $\\frac{4}{3}\\pi r^3=36\\pi$. Dividing both sides by $\\frac{4}{3}\\pi$ yields $r^3 = 27$; cube rooting both sides yields $r=3$. The surface area of a sphere with radius $r$ is $4\\pi r^2$; here, our surface area is $4\\pi(3^2)=\\boxed{36\\pi}$." ], "rewards": [ 1.0, 1.0, 0.0, 0.0, 1.0, 1.0 ] }