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 volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle 8 is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by S = 6sh - 25²cot 8 + (35²√3/2) csc 8 where s, the length of the sides of the hexagon, and h, the height, are constants. rear of cell (a) Calculate ds/de. de trihedral -angle 9 front of cell (b) What angle should the bees prefer?"."> (Round your answer to two decimal places.) • (c) Determine the minimum surface area of the cell (in terms of s and h).

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 volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle 8 is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by S = 6sh - 25²cot 8 + (35²√3/2) csc 8 where s, the length of the sides of the hexagon, and h, the height, are constants. rear of cell (a) Calculate ds/de. de trihedral -angle 9 front of cell (b) What angle should the bees prefer?"."> (Round your answer to two decimal places.) • (c) Determine the minimum surface area of the cell (in terms of s and h).