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?


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? $\endgroup$ – user2451412 Oct 4 '14 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! $\endgroup$ – Florian D'Souza Oct 4 '14 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.. $\endgroup$ – user2451412 Oct 4 '14 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. $\endgroup$ – user2451412 Oct 4 '14 at 13:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.