To find the correct relationship between the volumes of the sphere, cylinder, and cone when they share the same radius and where the height of the cylinder and cone is twice the radius, let's calculate each volume using the following formulas:
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Volume of the Sphere (V_sphere): \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Volume of the Cylinder (V_cylinder): The height of the cylinder (\(h\)) is twice the radius (\(h = 2r\)): \[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3 \]
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Volume of the Cone (V_cone): The height of the cone is also \(h = 2r\): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can summarize our volumes:
- \(V_{\text{sphere}} = \frac{4}{3} \pi r^3\)
- \(V_{\text{cylinder}} = 2 \pi r^3\)
- \(V_{\text{cone}} = \frac{2}{3} \pi r^3\)
Next, we check the relationships given in the options:
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Option: cylinder = sphere – cone \[ V_{\text{cylinder}} = V_{\text{sphere}} - V_{\text{cone}} \] Substituting the volumes gives: \[ 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 \] \[ 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This does not hold true.
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Option: sphere = cylinder + cone \[ V_{\text{sphere}} = V_{\text{cylinder}} + V_{\text{cone}} \] Substituting gives: \[ \frac{4}{3} \pi r^3 = 2 \pi r^3 + \frac{2}{3} \pi r^3 \] \[ \frac{4}{3} \pi r^3 \neq 2 \pi r^3 + \frac{2}{3} \pi r^3 \] This is also not true.
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Option: cone = sphere – cylinder \[ V_{\text{cone}} = V_{\text{sphere}} - V_{\text{cylinder}} \] Substituting: \[ \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 - 2 \pi r^3 \] \[ \frac{2}{3} \pi r^3 \neq \frac{4}{3} \pi r^3 - \frac{6}{3} \pi r^3 \] Not true.
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Option: cone = cylinder – sphere \[ V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}} \] Substituting: \[ \frac{2}{3} \pi r^3 = 2 \pi r^3 - \frac{4}{3} \pi r^3 \] \[ \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This holds true!
Thus, the correct relationship is:
cone = cylinder – sphere