Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 2 / 19 / a / Solution

Fix the total area at $A_0>0$. Apply Lagrange multipliers to $V=\pi a^2\ell$ with constraint $G=2\pi a(a+\ell )=A_0$. The two derivative equations areSince $a>0$, the second gives $\lambda =a/2$. Substitution in the first gives $\ell =2a$. The constraint then yields $a^2=A_0/(6\pi )$. ThereforeFor completeness, eliminating $\ell$ makes $V=(A_0/2)a-\pi a^3$ on $0<a<\sqrt{A_0/(2\pi )}$. Its derivative changes from positive to negative at the calculated radius and its volume tends to zero at both domain endpoints. Thus this stationary cylinder is indeed the maximum. It is the fixed-area case of optimal aspect ratios of closed cylinders.