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Problem 47 Hard Difficulty

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 $ \theta $ is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area $ S $ is given by
$$ S = 6sh - \dfrac{3}{2}s^2 \cot \theta + (3s^2\sqrt{3} /2) \csc \theta $$
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 $).
Note: Actual measurements of the angle $ \theta $ in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than $ 2^{\circ} $.

Answer

a) $\frac{3}{2} s^{2} \cos ^{2} \theta(1-\sqrt{3} \cos \theta)=0$ $\\$
b) $ 54.7^{\circ}$ $\\$
c) $6 s h+\frac{3 s^ {2}}{\sqrt{2}}$

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Video Transcript

So for this problem, we start off with this formula as equals six s h, minus three halves s squared co tangent theta plus Move this a little bit plus three s squared root 3/2 times co seeking theta. Then if we differentiate with respect to theta what we end up getting is DS. The theater equals three halves asked squared times, coasts again squared theta because we contingent derivative contingent coast to coast and square Data minus s minus three s squared Route three over to And then remember the coast. Second derivative is going to give us coast seeking theatre co tangent beta. So that's going to be our first answer for partner. Then we have port beam and for part B, we want to minimize the surface area over volume, so surface areas already given and volume will be independent of data. So that's because volume of a prism is one third height times the area of the base, so height an area of the base or independent of freedom. So what we need is data, um, for which surface area is minimum. So given what we already have, which is this equation up here we've already differentiated with respect to theta, and we got this and then we set that equal to zero. So if we take this right here instead of equal to zero, we end up getting one over root three equals co sign data. So theater is going to equal approximately. Yeah, 55 degrees. So now we've determined the data for which we can minimize, um, our volume, um, or minimize our surface area. So then we substitute this in for the given expression, So now we have s of 55 degrees and then plug that in. So what we end up getting as a result is going to be one state is in wind up getting the minimum surface area to be six s h plus three over root, two as squared. So this will be our final answer for a minimal surface area. Based on the fact that we just found