Let's analyze the volumes of the sphere, cylinder, and cone based on the given conditions.
Let the radius be \( r \) and the height of the cylinder and cone be \( h = 2r \).
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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): \[ V_c = \pi r^2 h = \pi r^2 (2r) = 2 \pi r^3 \]
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Volume of the Cone (V_co): \[ V_co = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now, let's compare these volumes to see which equations hold true:
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Three Cones Compared to One Cylinder: \[ 3 V_co = 3 \left(\frac{2}{3} \pi r^3\right) = 2 \pi r^3 = V_c \] \( 3 \) cones do equal \( 1 \) cylinder.
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Sphere vs. Cylinder and Cone: Combine the volumes of the cylinder and cone: \[ V_c + V_co = (2 \pi r^3) + \left(\frac{2}{3} \pi r^3\right) = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] This does not equal the sphere's volume.
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Sphere Minus Cone Compared to Cylinder: \[ V_s - V_co = \frac{4}{3} \pi r^3 - \left(\frac{2}{3} \pi r^3\right) = \frac{2}{3} \pi r^3 \] This does not equal the volume of the cylinder since \( V_c = 2 \pi r^3 \).
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Equation Relating Cylinder, Cones, and Sphere: We need to know whether: \[ 1 V_c = 2 V_co + V_s \] Since \( V_c = 2 \pi r^3, \) \( V_co = \frac{2}{3} \pi r^3, \) and \( V_s = \frac{4}{3} \pi r^3, \) we can rewrite this as: \[ 2V_co + V_s = 2\left(\frac{2}{3} \pi r^3\right) + \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 + \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \] This does not equal \( V_c \).
Conclusion
The relationship that holds true is: 3 cones = 1 cylinder.
This is the correct response.