Question

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 is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by

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

(a) Calculate .

(b) What angle should the bees prefer?

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