Let's analyze the volume formulas for a cylinder, sphere, and cone.
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Cylinder: The formula for the volume \( V \) of a cylinder is given by: \[ V_{\text{cylinder}} = \pi r^2 h \] Given that the height \( h \) is twice the radius \( r \) (i.e., \( h = 2r \)), we can substitute this into the formula: \[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]
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Sphere: The formula for the volume \( V \) of a sphere is: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
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Cone: The formula for the volume \( V \) of a cone is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Substituting \( h = 2r \), we get: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now we have the volumes:
- Volume of the cylinder: \( V_{\text{cylinder}} = 2\pi r^3 \)
- Volume of the sphere: \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)
- Volume of the cone: \( V_{\text{cone}} = \frac{2}{3} \pi r^3 \)
Relationship Between Volumes
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The volume of the cylinder is 3 times the volume of the cone: \[ V_{\text{cylinder}} = 3 \cdot V_{\text{cone}} \quad \text{(since } 2\pi r^3 = 3 \cdot \frac{2}{3} \pi r^3\text{)} \]
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The volume of the sphere is actually equal to \( \frac{2}{3} \) of the volume of the cylinder: \[ V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}} \quad \text{(since } \frac{4}{3} \pi r^3 = \frac{2}{3} (2\pi r^3)\text{)} \]
Summary
- Cylinder volume = 3 × Cone volume.
- Sphere volume = \( \frac{2}{3} \) × Cylinder volume.
These relationships show how the volumes of these shapes relate to each other given the conditions of the problem.