A cone, a hemisphere and a cylinder stand on equal base of radius r and have the same height. If the sum of volumes of cone, the hemisphere and the cylinder is equal to volume of a sphere of radius R, then what is \frac{R^3}{r^3} equal to ?

Explanation

For the hemisphere, height h=r, so h=r for all three. The sum of volumes is \frac{1}{3}\pi r^2(r) + \frac{2}{3}\pi r^3 + \pi r^2(r) = 2\pi r^3. The volume of the sphere is \frac{4}{3}\pi R^3. Equating them gives \frac{4}{3}\pi R^3 = 2\pi r^3 \implies \frac{R^3}{r^3} = \frac{6}{4} = 1.5.

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