MY NOTES In a beehive, each cell is a regular hexagonal prism, open at one end; the other end is capped by three congruent rhombi forming a trihedral angle at the apex, as in the figure. Let 0 be the angle at which each rhombus meets the altitude, s the side length of the hexagon, and h the length of the longer base of the trapezoids on the sides of the cell. It can be shown that if s and h are held fixed, then the volume of the cell is constant (independent of 0), and for a given value of 0 the surface area S of the cell is S = 6sh - 2 52 cot(0) + 352 3 apex open end i It is believed that bees form their cells in such a way as to minimize surface area, thus using the least amount of wax in cell construction. (a) Calculate- ds de as de (b) What angle 0 should the bees prefer?t (Round your answer to two decimal places.) (c) Determine the minimum surface area of the cell (in terms of s and h). Note: Actual measurements of the angle 0 in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than 20. Need Help? Read It Watch It