# Surface area optimization of right cylinder and hemisphere

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.

• What do you mean by "the volume isn't given"? It is given and it's $V$. – 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 $r$.

• 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'? – Harp Jan 29 '17 at 6:17
• $V$ is constant (its derivative is zero), and there's no $h$ present! – 1Emax Jan 29 '17 at 7:02
• 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. – Leo Granger Jun 30 '19 at 9:24