?In a beehive, each cell is a regular hexagonal prism, open at one end; the other end is
Chapter 4, Problem 53(choose chapter or problem)
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 π³ be the angle at which each rhombus meets the altitude, π the side length of the hexagon, and β the length of the longer base of the trapezoids on the sides of the cell. It can be shown that if π and β are held fixed, then the volume of the cell is constant (independent of π³), and for a given value of π³ the surface area π of the cell is
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 ππ/ππ³.
(b) What angle π³ should the bees prefer?
(c) Determine the minimum surface area of the cell in terms of π and β.
Note:Β Actual measurements of the angle π³ in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than 28.
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