9)

rear trihedral of cell angle o front S of cell 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 @ is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by S = 6sh -$2 cot 0 + (3:2\\/3/2) cscewhere s, the length of the sides of the hexagon, and h, the height, are constants. (a) Calculate dS/de. (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)