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

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

where $ s $, the length of the sides of the hexagon, 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} $.