To analyze the relationship between the volumes of the sphere, cylinder, and cone, let's start by calculating their respective volumes using the given dimensions.
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Sphere: The volume \( V \) of a sphere is given by the formula: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Cylinder: The volume of a cylinder is given by: \[ V_{\text{cylinder}} = \pi r^2 h \] Given that the height \( h \) is twice the radius \( r \): \[ h = 2r \quad \Rightarrow \quad V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]
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Cone: The volume of a cone is given by: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Again, using \( h = 2r \): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, we can compare the volumes:
- Volume of the sphere: \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)
- Volume of the cylinder: \( V_{\text{cylinder}} = 2\pi r^3 \)
- Volume of the cone: \( V_{\text{cone}} = \frac{2}{3} \pi r^3 \)
Now, let's check the relationships between the volumes:
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Check \( V_{\text{cylinder}} = V_{\text{sphere}} - V_{\text{cone}} \): \[ V_{\text{sphere}} - V_{\text{cone}} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This is equal to \( V_{\text{cone}} \), not \( V_{\text{cylinder}} \).
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Check \( V_{\text{sphere}} = V_{\text{cylinder}} + V_{\text{cone}} \): \[ V_{\text{cylinder}} + V_{\text{cone}} = 2\pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] This does not equal \( V_{\text{sphere}} \).
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Check \( V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}} \): \[ V_{\text{cylinder}} - V_{\text{sphere}} = 2\pi r^3 - \frac{4}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This is equal to \( V_{\text{cone}} \).
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Check \( V_{\text{cone}} = V_{\text{sphere}} - V_{\text{cylinder}} \): \[ V_{\text{sphere}} - V_{\text{cylinder}} = \frac{4}{3} \pi r^3 - 2 \pi r^3 = \frac{4}{3} \pi r^3 - \frac{6}{3} \pi r^3 = -\frac{2}{3} \pi r^3 \] This does not equal \( V_{\text{cone}} \).
From this analysis:
The correct relationship is: \[ V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}} \]
Therefore, the correct response is: cone = cylinder − sphere.