Describe briefly the method of Lagrange multipliers for finding the stationary points of a function $f(x, y)$ subject to a constraint $\phi(x, y)=0$.

A tent manufacturer wants to maximize the volume of a new design of tent, subject only to a constant weight (which is directly proportional to the amount of fabric used). The models considered have either equilateral-triangular or semi-circular vertical crosssection, with vertical planar ends in both cases and with floors of the same fabric. Which shape maximizes the volume for a given area $A$ of fabric?

[Hint: $(2 \pi)^{-1 / 2} 3^{-3 / 4}(2+\pi)<1 .$ ]