Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 19 / a / Solution

A central cross-section of the sphere gives the constraint $r^2+h^2/4=R^2$, and the cylinder's volume is $V=\pi r^{2}h$. At an interior optimum with $r,h>0$, the Lagrange multiplier equations for this constrained optimization problem areThe first gives $\lambda =\pi h$; substitution into the second gives $r^2=h^2/2$. The constraint then givesThe feasible nonnegative $(r,h)$ form a compact arc, whose endpoints have zero volume. Since this is the only positive stationary candidate, it is the global maximum. Its volume and fraction of the sphere's volume are