In a beehive, each ell is a regular hexagonal prism, open at one end with a trihedal angle at the other end. It is believed that bees form their cells in such a way as to minimize the surface area for a given volume, 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 $dS/d \theta$. (Your answer may depend on s, h, and t (use t instead of theta)).

(b) What angle should bees prefer (in radians)?

(c) Determine the minimum surface area of the cell (Your answer may depend on s and h).