Bees build honeycomb structures out of cells with a hexagonal base and three

Chapter 4, Problem 80

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Bees build honeycomb structures out of cells with a hexagonal base and three rhombus-shaped faces on top, as in Figure 23. We can show that the surface area of this cell is A( ) = 6hs + 3 2 s2( 3 csc cot ) with h, s, and as indicated in the figure. Remarkably, bees know which angle minimizes the surface area (and therefore requires the least amount of wax). (a) Show that 54.7 (assume h and s are constant). Hint: Find the critical point of A( ) for 0 < < /2. (b) Confirm, by graphing f ( ) = 3 csc cot , that the critical point indeed minimizes the surface area. s h FIGURE 23 A cell in a honeycomb constructed by bees.