To determine the relationship between the volumes of the sphere, cylinder, and cone, let's first calculate the volumes of each shape using the given dimensions.
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Volume of the Sphere (V_s): \[ V_s = \frac{4}{3} \pi r^3 \]
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Volume of the Cylinder (V_c): The cylinder has a radius \( r \) and a height of \( 2r \). \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the Cone (V_co): The cone also has a radius \( r \) and a height of \( 2r \). \[ V_co = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, let's summarize the volumes:
- Sphere: \( V_s = \frac{4}{3} \pi r^3 \)
- Cylinder: \( V_c = 2 \pi r^3 \)
- Cone: \( V_co = \frac{2}{3} \pi r^3 \)
Now we'll compute relationships between these volumes.
- Cylinder vs. Sphere and Cone:
- \( V_c = 2\pi r^3 \)
- \( V_s - V_co = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \) (which is equal to \( V_co \))
Thus, \[ V_c = V_s - V_co \]
- Cone vs. Cylinder and Sphere:
- \( V_co = \frac{2}{3} \pi r^3 \)
- \( V_c - V_s = 2\pi r^3 - \frac{4}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \) (which is equal to \( V_co \))
So, \[ V_co = V_c - V_s \]
- Sphere vs. Cylinder and Cone:
- \( V_s = \frac{4}{3} \pi r^3 \)
- \( V_c + V_co = 2\pi r^3 + \frac{2}{3} \pi r^3 \)
- Converting \( 2\pi r^3 \) into thirds gives \( \frac{6}{3} \pi r^3 \): \[ V_c + V_co = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{6 + 2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]
This means: \[ V_s < V_c + V_co \quad (since V_s = \frac{4}{3} \pi r^3 \text{ and } V_c + V_co = \frac{8}{3} \pi r^3) \]
Based on the calculations above, the only relation that holds true is:
\[ \text{Cylinder} = \text{Sphere} - \text{Cone} \]
Therefore, the correct option is: cylinder = sphere – cone.