A sphere is inscribed inside a pyramid with a square as a base whose height is $15$ times the length of one edge of the base. A cube is inscribed inside the sphere. What is the ratio of the volume of the pyramid to the volume of the cube?

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Question The problem never mentions the side lengths of the pyramid, only the height. Therefore how can the solution conclude that the plane cut is isosceles?

  • $\begingroup$ Is it because the pyramid is inscribable with a sphere? That leads to the question of where it is possible to inscribe a sphere inside a square pyramid that isn't equilateral? $\endgroup$ – John Ryan Dec 23 '15 at 19:21

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