0
$\begingroup$

How do I solve this calculus problem:

A farm is trying to build a metal silo with volume V. It consists of a hemisphere placed on top of a right cylinder. What is the radius which will minimize the construction cost (surface area).

I'm not sure how to solve this problem as I can't substitute the height when the volume isn't given.

| cite | improve this question | | | | |
$\endgroup$
  • 1
    $\begingroup$ What do you mean by "the volume isn't given"? It is given and it's $V$. $\endgroup$ – 1Emax Jan 29 '17 at 6:00
3
$\begingroup$

Let's call the radius $r$, the height $h$, and the surface $S$. Then $$\tag {1} V = \pi r^2h+\frac{2}{3}\pi r^3,$$ and $$S = 2\pi r h + 2 \pi r^2=2\pi r(h+r).$$

Substituting $h$ from $(1)$ we get $$\tag{2} S = 2 \pi r (\frac{V}{\pi r^2}+\frac{1}{3}r).$$

Now all you have to do is minimize $(2)$ with respect to $r$.

| cite | improve this answer | | | | |
$\endgroup$
  • $\begingroup$ After differentiating I will still be left with V'. After differentiating V I'm stuck with h'. Can I still find the intercepts of S' when it contains r and h'? $\endgroup$ – Harp Jan 29 '17 at 6:17
  • $\begingroup$ $V$ is constant (its derivative is zero), and there's no $h$ present! $\endgroup$ – 1Emax Jan 29 '17 at 7:02
  • $\begingroup$ This guy is plain wrong. The surface area is $$3 \pi r^2 + 2 \pi rh .$$ Therefore anything that comes after that is naturally wrong. Please check your answer before posting. $\endgroup$ – Leo Granger Jun 30 '19 at 9:24

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.