Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 11 / c / i / Solution

An axial section gives similar right triangles. At height $x$ above the base, the cone's remaining height is $h-x$, so the cylinder's radius is $r=(h-x)\tan \alpha$. Therefore its volume isAssume $h>0$ and $0<\alpha <\pi /2$ for a nondegenerate cone. The geometry and the dimensionless optimization curves are illustrated below.