To analyze the relationship between the volumes of a sphere, cylinder, and cone, we can start by calculating the volumes based on the given dimensions:
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Definitions:
- Let the radius \( r \) be the same for all shapes.
- The height \( h \) of the cylinder and cone is \( 2r \).
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Volume Formulas:
- Volume of the sphere (\( V_s \)): \[ V_s = \frac{4}{3} \pi r^3 \]
- Volume of the cylinder (\( V_c \)): \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
- Volume of the cone (\( V_{cone} \)): \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
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Now, we can examine the relationships:
- 1. sphere - cone = cylinder: \[ V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \quad \text{(not equal to } V_c) \]
- 2. 3 cones = 1 cylinder: \[ 3 V_{cone} = 3 \cdot \left(\frac{2}{3} \pi r^3\right) = 2 \pi r^3 = V_c \quad \text{(this is true)} \]
- 3. sphere = cylinder + cone: \[ V_s = V_c + V_{cone} \Rightarrow \frac{4}{3} \pi r^3 = 2\pi r^3 + \frac{2}{3} \pi r^3 \quad \text{(not true)} \]
- 4. 1 cylinder = 2 cones + 1 sphere: \[ V_c = 2 V_{cone} + V_s \Rightarrow 2 V_{cone} + V_s = 2 \cdot \frac{2}{3} \pi r^3 + \frac{4}{3} \pi r^3 \Rightarrow \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \quad \text{(not true)} \]
The only true statement is:
3 cones = 1 cylinder.