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Cylinder inside of a Cube : Ratio of Volumes

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A cylinder is inscribed in a cube. That cylinder has a sphere inscribed inside of it. What is the ratio of the volume of the sphere to the volume of the cube?
HINT: Let X be a side of a square on the cube.

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The ratio of volumes of a cylinder inside a cube is calculated. The solution is detailed and well presented. the solution includes a diagram.

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