Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 19 / b / Solution

The base lies in the plane $z=R-h$. Its circular rim lies on the sphere, soMaximize the cone volume $V_c=\pi r^{2}h/3$ using the Lagrange multiplier expressionFor a nondegenerate cone, the $r$ equation gives $\lambda =\pi h/3$. The $h$ equation is $\pi r^2/3+2\lambda (R-h)=0$, hence $r^2=2h(h-R)$. Combining this with $r^2=2Rh-h^2$ givesIndeed, eliminating $r$ gives $V_c(h)=\pi h^2(2R-h)/3$, which vanishes at the endpoints and has a unique positive interior maximum at $4R/3$. Thus