I'm solving a problem involving calculus optimization. The problem is the following:

"We plan to build a boxshaped shelter with no floor and one side open. (Hence we need a roof and three sides). The base of the shelter is a square. The volume of the shelter is 144 cubic meters. Find the dimensions of the shelter that minimizes the material, i.e. the surface area. Use differentiation."

Everything I say from here on out are my conclusions:

I'm assuming that the shelter is a rectangular prism. Since the base of the shelter is a square, I assume the roof is as well.

So since the roof is a square, and a square's sides are all of equal length, that means that the width/length of the roof is really the same variable. So let x = length = width. Let y denote the height.

We want to minimize the surface area of the rectangle. The surface area of a rectangle is as follows:

$$SA = 2LW + 2HW + 2LH$$

However, since we are missing the bottom of the rectangle, $L*W$, and one of the sides, either $L*W$ or $H*W$, our SA formula becomes:

$$SA = x^2 + yx + 2xy$$ or $$SA = x^2 + 3xy$$

Remember that $x = length = width$ and $y = height$.

We have more than one variable in that equation, let us connect the two. We're given that the volume of the shelter is 144. So:

$$144 = x^2 * y$$ $$y = \frac{144}{x^2}$$

And this means that:

$$SA = x^2 + \frac{432}{x}$$

Now we differentiate the function and attempt to find the critical points (where $f'(x) = 0$ or $f'(x)$ is undefined. I.e. we want the local minimum..

Differentiating SA, it becomes:

$$SA' = 2x - 432x^{-2}$$

Is this really correct or did I go very wrong somewhere along the way? If by some miracle this is correct, how would I go about finding the local minimum?


1 Answer 1


This looks about right to me. You would need to set the 1st derivative you found to zero, and find the critical value. To make sure it is a minimum, you would then calculate the 2nd derivative. If the 2nd derivative with the critical value put in is positive, then you have a minimum.

  • $\begingroup$ Thanks, this helped. One thing about the first derivative though, I find it very hard to see when it's zero. I tried to factor it as follows: $SA' = 2x(1 - 216x^{-3})$. Seeing this, I know if x is zero the function is undefined. So presumably $216x^{-3}$ must become one, but how can I find this out in an easy way? $\endgroup$ Oct 4, 2014 at 13:29
  • $\begingroup$ The easiest way for me is to realize that x=0 is not physically possible. You can't build anything if one (or more) of your sides has a length/width/height of zero! $\endgroup$ Oct 4, 2014 at 13:31
  • $\begingroup$ Alright so we know $x \neq 0$, so when does $2x - 432x^(-2)$ become zero? (In order to find the local minimum). Sorry if I'm missing something.. $\endgroup$ Oct 4, 2014 at 13:37
  • $\begingroup$ I think I got it through Algebra. x = 6. I should be able to find y through the volume formula. Thank you for your help. Marking yours as the answer. $\endgroup$ Oct 4, 2014 at 13:48

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