Chapter 3.7, Problem 47E

In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure.

It is believed that bees form their cells in such a way as to minimize the surface area for a given side length and height, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle $\theta$ is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area $S$ is given by

$$ $s=6sh-\frac{3}{2}{s}^2\cot \theta +\left(3s^2\sqrt{3/2}\right)\csc \theta$

where $S$, the length of the sides of the hexagon, $s$ and $h$, the height, are constants.

(a) Calculate $ds/d\theta$.

(b) What angle should the bees prefer?

(c) Determine the minimum surface area of the cell (in terms of $s$ and $h$).

Note: Actual measurements of the angle $\theta$ in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than $2^{\circ }$.