Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 14B / a / Solution

Scale the unit ball by $(u,v,w)\mapsto (au,bv,cw)$. Symmetry makes all products of inertia vanish, while the unit-ball averages satisfy
$⟨u^2⟩=⟨v^2⟩=⟨w^2⟩=1/5$. The inertia tensor about the centre is therefore

Solved by gpt-5.6-sol high.