a) Differentiate with regards to .
b) Find the angle that gives the smallest surface area (use the result from a).
c) Use the result from b.
In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end. 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 each cell construction. Examination of these cells has shown that the measure of the apex angle theta is amazingly consistent. Based on the geometry of the cell it can be shown that the surface area S is given by
S=6sh-(3/2)s^2*cot(theta)+[(3s^2*(3)^.5)/2]*csc(theta)
where s, the length of the sides of the hexagon, and h, the height, are constants.
a) Calculate ds/d(theta)
b) What angle should the bees prefer?
c) Determine the minimum surface area of the cell in terms of s and h.
That looks correct. You can also check your answer here: webMathematica Explorations: Step-by-Step Derivatives