To analyze the relationships between the volumes of the sphere, cylinder, and cone, we can use their respective volume formulas:
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Sphere: The volume \( V_{\text{sphere}} \) of a sphere is given by: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Cylinder: The volume \( V_{\text{cylinder}} \) of a cylinder is given by: \[ V_{\text{cylinder}} = \pi r^2 h \] Given that the height \( h \) of the cylinder is twice the radius \( r \) (i.e., \( h = 2r \)), we can substitute to find: \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]
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Cone: The volume \( V_{\text{cone}} \) of a cone is given by: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Similarly, since the height \( h \) of the cone is also \( 2r \): \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we can compare the volumes we've calculated:
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Sphere volume: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Cylinder volume: \[ V_{\text{cylinder}} = 2\pi r^3 \]
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Cone volume: \[ V_{\text{cone}} = \frac{2}{3} \pi r^3 \]
Next, we can sum the volumes of the cylinder and the cone: \[ V_{\text{cylinder}} + V_{\text{cone}} = 2\pi r^3 + \frac{2}{3} \pi r^3 \] To combine these volumes, we convert \( 2\pi r^3 \) to a fraction with a denominator of 3: \[ 2\pi r^3 = \frac{6}{3} \pi r^3 \] Adding them together: \[ V_{\text{cylinder}} + V_{\text{cone}} = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]
Now we will compare \( \frac{8}{3} \pi r^3 \) to \( V_{\text{sphere}} \): \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \] We see that \( V_{\text{cylinder}} + V_{\text{cone}} = \frac{8}{3} \pi r^3 \neq \frac{4}{3} \pi r^3 \), so the first relationship does not hold.
Next, let's explore if the cone \( = \text{cylinder} - \text{sphere} \): To check this, we can calculate \( V_{\text{cylinder}} - V_{\text{sphere}} \): \[ V_{\text{cylinder}} - V_{\text{sphere}} = 2\pi r^3 - \frac{4}{3} \pi r^3 \] Converting \( 2\pi r^3 \) to a fraction with a denominator of 3: \[ 2\pi r^3 = \frac{6}{3} \pi r^3 \] Then: \[ V_{\text{cylinder}} - V_{\text{sphere}} = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 = V_{\text{cone}} \] This confirms that: \[ V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}} \]
Thus, we conclude that the correct relationship among the options given is:
cone = cylinder – sphere