To compare the volumes of the sphere, cylinder, and cone, we need to use their volume formulas and the given relationships.
Let's denote the radius as \( r \). The height of the cylinder and cone is given as \( h = 2r \).
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Volume of the sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Volume of the cylinder: \[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now let's examine the relationships:
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Relationship between cylinders and cone: \[ 2 V_{\text{cone}} = 2 \left( \frac{2}{3} \pi r^3 \right) = \frac{4}{3} \pi r^3 = V_{\text{sphere}} \quad \text{(so, this is not true)} \]
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Relationship between cones and sphere: \[ 2 V_{\text{cone}} = \frac{4}{3} \pi r^3 = V_{\text{sphere}} \quad \text{(this is true)} \]
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Relationship between cylinders and cones: Let's calculate the volume of 2 cones: \[ 2 V_{\text{cone}} = 2 \left( \frac{2}{3} \pi r^3 \right) = \frac{4}{3} \pi r^3 \] We already established that \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \).
Now, let's check how many cones fit in a cylinder: \[ V_{\text{cylinder}} = 2 \pi r^3 \quad \text{and} \quad 3 V_{\text{cone}} = 3 \left( \frac{2}{3} \pi r^3 \right) = 2 \pi r^3 \quad \text{(this is true)} \]
Thus:
- \( 2 V_{\text{cone}} = V_{\text{sphere}} \)
- \( 3 V_{\text{cone}} = V_{\text{cylinder}} \)
So the only true statement is:
2 cones = 1 sphere.
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