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.
Best Answer
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 $r$.