id stringclasses 2
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values | image imagewidth (px) 483 545 | subquestions listlengths 5 5 | final_question stringclasses 2
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49 | A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
<image1> | [
{
"answer": "(4/3)蟺r^3",
"question": "What is the formula for the volume of a sphere with radius r?"
},
{
"answer": "(1/3)蟺h(R^2 + Rr + r^2)",
"question": "What is the formula for the volume of a truncated cone with bottom radius R, top radius r, and height h?"
},
{
"answer": "(1/3)蟺h(R^... | If the volume of the outer transparent container is double the volume of the inner solid object, what is the ratio of the container's bottom radius to its top radius? | Let R be the bottom radius and r_t be the top radius of the truncated cone. Let r_s be the radius of the inscribed sphere. From the visual arrangement (sphere inscribed in a truncated cone), two key geometric relationships can be derived:
1. The height of the cone `h` is equal to the diameter of the sphere: `h = 2*r_s`... | |
18 | What is the maximum value of y? | [
{
"answer": "x^2 + y^2 = 25",
"question": "What is the equation representing the circle in the image?"
},
{
"answer": "5",
"question": "What is the radius of the circle?"
},
{
"answer": "(0, 0)",
"question": "Where is the center of the circle located?"
},
{
"answer": "(0, 5)"... | Calculate the y-intercept of the line that is tangent to the circle at the labeled point. | 1. From the image, the circle's equation is x虏 + y虏 = 25, which means its center is at the origin (0, 0).
2. The labeled point of tangency is (3, -4).
3. The slope of the radius connecting the center (0, 0) to the point of tangency (3, -4) is m_radius = (-4 - 0) / (3 - 0) = -4/3.
4. The tangent line is perpendicula... |
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