Chapter 4.1, Problem 73E

Honeycomb The surface area of a cell in a honeycomb is

$S=6hs+\frac{3s^2}{2}\left(\frac{\sqrt{3}-\cos \theta }{\sin \theta }\right)$

where h and s are positive constants and is the angle at which the upper faces meet the altitude of the cell (see figure). Find the angle $\theta (\pi /6\le \theta \le \pi /2)$ that minimizes the surface area S.