Honeycomb
The surface area of a cell in a honeycomb is
where h and s are positive constants and is the angle at which the upper faces meet the altitude of the cell (see figure). Find the angle that minimizes the surface area S.
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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 figure.
It is believed that bees form their cells in such a way as to minimize the surface area for a given side length and height, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle isĀ Ā is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given byĀ
Ā Ā
Where S, the length of the sides of the hexagon, s and h, the height are constants.
(a) CalculateĀ Ā Ā
(b) What angle should the bees prefer?
(c) Determine the minimum surface area of the cell (in terms of s and h)
Note: Actual measurements of the angleĀ Ā in beehives have been made, and the measures of these angles seldom differ from the calculated value by more thanĀ Ā Ā
Ā
It is believed that bees form their cells in such a way as to minimize the surface area for a given side length and height, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle isĀ is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given byĀ
(a) Calculate Ā
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 Ā
Verifying a Formula
(a) Given a circular sector with radius L and central angle (see figure), show that the area of the sector is given byĀ .
(b) By joining the straight-line edges of the sector in part (a), a right circular cone is formed (see figure) and the lateral surface area of the cone is the same as the area of the sector. Show that the area is , where r is the radius of the base of the cone. (Hint: The arc length of the sector equals the circumference of the base of the cone.)
(c) Use the result of part (b) to verify that the formula for the lateral surface area of the frustum of a cone with slant height L and radii r1 and r2 (see figure) is .(Note: This formula was used to develop the integral for finding the surface area of a surface of revolution.)
