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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.

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    What do you mean by "the volume isn't given"? It is given and it's $V$. – 2017-01-29

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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 $r$.

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    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'? – 2017-01-29
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    $V$ is constant (its derivative is zero), and there's no $h$ present! – 2017-01-29