ILL 1974 Problem 5

A straight cone is given inside a rectangular parallelepiped $B$ , with the apex at one of the vertices, say $T$ , of the parallelepiped, and the base touching the three faces opposite to $T .$ Its axis lies at the long diagonal through $T.$ If $V_1$ and $V_2$ are the volumes of the cone and the parallelepiped respectively, prove that $$V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.$$