In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral 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 θ is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area is given by S = 6sh €“ 3/2s2cot θ + (3s2√3/2) csc θ where s, the length of the sides of the hexagon, and h, the height, are constants.
(a) Calculate dS/dθ.
(b) What angle should the bees prefer?
(c) Determine the minimum surface area of the cell (in terms of and h).
Note: Actual measurements of the angle θ in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than 2o.

Transcribed Image Text:

trihedral rear angle e of cell front of cell