To find the correct relationship between the volumes of a sphere, cylinder, and cone, we first need to calculate the volumes of each shape using their respective formulas.
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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 height (h) of the cylinder is twice the radius (r), so \( h = 2r \). \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \]
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Volume of the Cone (V_cone): The height (h) of the cone is also \( 2r \) (same as the height of the cylinder). \[ V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Next, we compare the volumes:
- Three cones: \[ 3 V_{cone} = 3 \left( \frac{2}{3} \pi r^3 \right) = 2 \pi r^3 \]
Since the volume of the cylinder is also \( 2 \pi r^3 \), we find: \[ 3 V_{cone} = V_c \] So, \( 3 \text{ cones} = 1 \text{ cylinder} \).
- Comparing the sphere and the cylinder: \[ V_s = \frac{4}{3} \pi r^3 \quad \text{and} \quad V_c = 2\pi r^3 \]
To check if the sphere is equal to the cylinder plus the cone: \[ V_c + V_{cone} = 2\pi r^3 + \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \quad (not equal to V_s) \]
Now let's check:
- Sphere minus cone: \[ V_s - V_{cone} = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \quad (which is not equal to V_c) \]
Given the calculations, we conclude that the correct relationship is: \[ \boxed{3 \text{ cones} = 1 \text{ cylinder}} \]
This statement is true and reflects the relationship accurately among the three volumes based on the given dimensions.
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