In a system where you have a sphere inscribed within a right circular cylinder and another sphere that circumscribes this cylinder, if the sphere inscribed in the cylinder has a radius r, find the ratio of the volume of the larger sphere to the volume of

The sphere inscribed in the cylinder has a radius r and thus a volume of (4/3)πr^3. The larger sphere circumscribing the cylinder has a diameter equal to the cylinder’s diagonal, which is 2√2r, giving it a radius of √2r and a volume of (4/3)π(√2r)^3. The ratio of the volumes is (√2)^3 = 2√2.

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