{ "log_file": [ "step_102-b_0.log", "step_102-b_1.log", "step_102-b_2.log", "step_102-b_3.log", "step_102-b_4.log", "step_102-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:\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### 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:\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### Response:\nSEARCH[\"\\\\cos(288\\\\pi)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/precalculus/1086.json\n\n#### Similar Question\nFind the number of solutions to the equation\n\\[\\tan (5 \\pi \\cos \\theta) = \\cot (5 \\pi \\sin \\theta)\\]where $\\theta \\in (0, 2 \\pi).$\n\n#### User Answer\nFrom the given equation,\n\\[\\tan (5 \\pi \\cos \\theta) = \\frac{1}{\\tan (5 \\pi \\sin \\theta)},\\]so $\\tan (5 \\pi \\cos \\theta) \\tan (5 \\pi \\sin \\theta) = 1.$\n\nThen from the angle addition formula,\n\\begin{align*}\n\\cot (5 \\pi \\cos \\theta + 5 \\pi \\sin \\theta) &= \\frac{1}{\\tan (5 \\pi \\cos \\theta + 5 \\pi \\sin \\theta)} \\\\\n&= \\frac{1 - \\tan (5 \\pi \\cos \\theta) \\tan (5 \\pi \\sin \\theta)}{\\tan (5 \\pi \\cos \\theta) + \\tan (5 \\pi \\sin \\theta)} \\\\\n&= 0.\n\\end{align*}Hence, $5 \\pi \\cos \\theta + 5 \\pi \\sin \\theta$ must be an odd multiple of $\\frac{\\pi}{2}.$ In other words,\n\\[5 \\pi \\cos \\theta + 5 \\pi \\sin \\theta = (2n + 1) \\cdot \\frac{\\pi}{2}\\]for some integer $n.$ Then\n\\[\\cos \\theta + \\sin \\theta = \\frac{2n + 1}{10}.\\]Using the angle addition formula, we can write\n\\begin{align*}\n\\cos \\theta + \\sin \\theta &= \\sqrt{2} \\left( \\frac{1}{\\sqrt{2}} \\cos \\theta + \\frac{1}{\\sqrt{2}} \\sin \\theta \\right) \\\\\n&= \\sqrt{2} \\left( \\sin \\frac{\\pi}{4} \\cos \\theta + \\cos \\frac{\\pi}{4} \\sin \\theta \\right) \\\\\n&= \\sqrt{2} \\sin \\left( \\theta + \\frac{\\pi}{4} \\right).\n\\end{align*}so\n\\[\\sin \\left( \\theta + \\frac{\\pi}{4} \\right) = \\frac{2n + 1}{10 \\sqrt{2}}.\\]Thus, we need\n\\[\\left| \\frac{2n + 1}{10 \\sqrt{2}} \\right| \\le 1.\\]The integers $n$ that work are $-7,$ $-6,$ $-5,$ $\\dots,$ $6,$ giving us a total of 14 possible values of $n.$ Furthermore, for each such value of $n,$ the equation\n\\[\\sin \\left( \\theta + \\frac{\\pi}{4} \\right) = \\frac{2n + 1}{10 \\sqrt{2}}.\\]has exactly two solutions in $\\theta.$ Therefore, there are a total of $28$ solutions $\\theta.$\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:\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### 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:\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### Response:\nSEARCH[\"\\\\cos(288\\\\pi)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/222.json\n\n#### Similar Question\nCircle $A$ has its center at $A(4, 4)$ and has a radius of 4 units. Circle $B$ has its center at $B(12, 4)$ and has a radius of 4 units. What is the area of the gray region bound by the circles and the $x$-axis? Express your answer in terms of $\\pi$. [asy]\nimport olympiad; size(150); defaultpen(linewidth(0.8));\nxaxis(0,16,Ticks(\"%\",1.0));\nyaxis(0,8,Ticks(\"%\",1.0));\nfill((4,4)--(12,4)--(12,0)--(4,0)--cycle,gray(0.7));\nfilldraw(circle((4,4),4),fillpen=white);\nfilldraw(circle((12,4),4),fillpen=white);\ndot(\"$A$\",(4,4),S); dot(\"$B$\",(12,4),S);\n[/asy]\n\n#### User Answer\nDraw a 4 by 8 rectangle with the vertices at $(4, 4), (12, 4), (12, 0)$ and $(4, 0)$. The area of that box is $4 \\times 8 = 32$ square units. From that we can subtract the area of the sectors of the 2 circles that are binding our shaded region. The area of each sector is $(1/4)4^2\\pi = 4\\pi$; therefore, we need to subtract $2(4\\pi) = 8\\pi$. This gives us $32 - 8\\pi$ square units.\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:\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### 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:\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### Response:\nSEARCH[\"\\\\cos(288\\\\pi)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/1041.json\n\n#### Similar Question\nIn the diagram, $O$ is the center of a circle with radii $OP=OQ=5$. What is the perimeter of the shaded region?\n\n[asy]\nsize(100);\nimport graph;\nlabel(\"$P$\",(-1,0),W); label(\"$O$\",(0,0),NE); label(\"$Q$\",(0,-1),S);\n\nfill(Arc((0,0),1,-90,180)--cycle,mediumgray);\ndraw(Arc((0,0),1,-90,180));\nfill((0,0)--(-1,0)--(0,-1)--cycle,white);\ndraw((-1,0)--(0,0)--(0,-1));\ndraw((-.1,0)--(-.1,-.1)--(0,-.1));\n\n[/asy]\n\n#### User Answer\nThe perimeter of the shaded region equals the sum of the lengths of $OP$ and $OQ$ plus the length of arc $PQ$.\n\nEach of $OP$ and $OQ$ has length 5.\n\nArc $PQ$ forms $\\frac{3}{4}$ of the circle with center $O$ and radius 5, because the missing portion corresponds to a central angle of $90^\\circ$, and so is $\\frac{1}{4}$ of the total circle.\n\nThus, the length of arc $PQ$ is $\\frac{3}{4}$ of the circumference of this circle, or $\\frac{3}{4}(2\\pi(5))=\\frac{15}{2}\\pi$. Therefore, the perimeter is $5+5+\\frac{15}{2}\\pi = 10 + \\frac{15}{2}\\pi$.\n\n--- RESULTS END ---\n\n" ], "response": [ " SEARCH[\"\\\\cos(288\\\\pi)\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nLet $r$ be the radius of the larger sphere, and let $R$ be the radius of the smaller sphere. Since the volume of a sphere is $\\frac{4}{3}\\pi r^3$, we have\n$$\\frac{4}{3}\\pi r^3 = \\frac{12.5}{100}\\left(\\frac{4}{3}\\pi r^3\\right) \\Longrightarrow r = \\frac{R}{2}.$$Thus, the ratio of the radius of the smaller sphere to the radius of the larger sphere is $\\boxed{\\frac{1}{2}}$.", " SEARCH[\"\\\\cos(288\\\\pi)\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nLet $r$ be the radius of the larger sphere, and let $s$ be the radius of the smaller sphere. We know that the volume of a sphere is $\\frac{4}{3}\\pi r^3$, so we have\n\n$$\\frac{4}{3}\\pi r^3 = \\frac{12.5}{100}\\left(\\frac{4}{3}\\pi r^3\\right) \\Longrightarrow \\frac{4}{3}\\pi r^3 = \\frac{5}{8}\\pi r^3 \\Longrightarrow \\frac{5}{8} = \\frac{r^3}{r^3} \\Longrightarrow r^2 = \\boxed{\\frac{5}{2}}.$$\n\nWe can check that this answer makes the volume of the smaller sphere equal to one-fourth the volume of the larger sphere:\n\n$$\\frac{4}{3}\\pi s^3 = \\frac{1}{4} \\left(\\frac{4}{3}\\pi r^3\\right) \\Longrightarrow \\frac{4}{3}\\pi s^3 = \\frac{1}{12}\\pi r^3 \\Longrightarrow s^3 = \\boxed{\\frac{1}{3}}r^3.$$", " SEARCH[\"\\\\cos(288\\\\pi)\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nLet $r$ be the radius of the larger sphere. Then the volume of the larger sphere is $\\frac{4}{3}\\pi r^3$. Let $s$ be the radius of the smaller sphere. Then the volume of the smaller sphere is $\\frac{4}{3}\\pi s^3$. Since the volume of the smaller sphere is 12.5\\% of the volume of the larger sphere, we have \\[\\frac{4}{3}\\pi s^3=\\frac{1}{8}\\cdot\\frac{4}{3}\\pi r^3.\\] Dividing both sides by $\\frac{4}{3}\\pi r^3$, we get \\[s^3=\\frac{1}{8}r^3.\\] Taking the cube root of both sides, we get \\[s=\\frac{1}{2}r.\\] Therefore, the ratio of $s$ to $r$ is $\\boxed{\\frac{1}{2}}$." ], "src_path": [ "train/geometry/698.json", "train/geometry/698.json", "train/geometry/698.json", "train/geometry/698.json", "train/geometry/698.json", "train/geometry/698.json" ], "input": [ "A 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.", "A 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.", "A 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.", "A 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.", "A 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.", "A 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." ], "output": [ "We 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 $\\boxed{\\frac{1}{2}}$.\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.)", "We 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 $\\boxed{\\frac{1}{2}}$.\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.)", "We 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 $\\boxed{\\frac{1}{2}}$.\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.)", "We 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 $\\boxed{\\frac{1}{2}}$.\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.)", "We 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 $\\boxed{\\frac{1}{2}}$.\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.)", "We 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 $\\boxed{\\frac{1}{2}}$.\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.)" ], "rewards": [ 1.0, 1.0, 0.0, 0.0, 1.0, 1.0 ] }