To analyze the volumes of the sphere, cylinder, and cone, we can use their volume formulas and the given information about their 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): Given the height (h) of the cylinder is twice the radius (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_co): The height of the cone is also 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 look at the relationships:
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3 cones: \[ 3 \times V_co = 3 \times \frac{2}{3} \pi r^3 = 2\pi r^3 = V_c \] This means 3 cones = 1 cylinder.
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Sphere in terms of cylinder and cone: \[ V_s = \frac{4}{3} \pi r^3 = V_c - V_co = 2\pi r^3 - \frac{2}{3} \pi r^3 = \frac{6}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 \] This means sphere = cylinder - cone.
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Sphere minus cone: \[ V_s - V_co = \frac{4}{3} \pi r^3 - \frac{2}{3} \pi r^3 = \frac{2}{3} \pi r^3 \] This is equal to \(\frac{1}{3} V_c\), not equal to the cylinder's volume.
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1 cylinder and cones and sphere: \[ V_c = 3 V_co + V_s = 2\pi r^3 \] This doesn't hold as well.
Thus, the relationship that is true is:
3 cones = 1 cylinder.