16 Nov 2021
Problem 33b
Page 307
Section 4.6: Optimization Problems
Chapter 4: Applications of Differentiation
Textbook ExpertVerified Tutor
16 Nov 2021
Given information
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 is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area is given by
where , the length of the sides of the hexagon, and , the height, are constants.
Step-by-step explanation
Step 1.
According to the question,
Given surface area is
Differentiating with respect to we derived in 33a