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.

  • 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

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 $r$.

  • $\begingroup$ 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. $\endgroup$ Jun 30 '19 at 9:24

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