Solution: In a beehive, each cell is a regular hexagonal prism, open at one end with a

Chapter 4, Problem 33

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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 gure. 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 is given byS 6sh 3 2s2 cot (3s2s3 2) csc where , the length of the sides of the hexagon, and , 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 and ). Note:Actual measurements of the angle in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than .