# Thread: How do I optimize surface area with set volume?

1. ## How do I optimize surface area with set volume?

I'm so confused right now. I don't know calculus, and that is how people have been explaining things to me as of yet, so if there is any other way of doing this can someone explain it?

You are to design a cylindrical tank with a minimum capacity of 300,000 cubic feet. Minimize the surface area of the tank, while keeping the set volume.

Um, help?

2. It can be shown that the surface area of a cylinder is minimum when height=diameter.

Calculus is one of the easiest to do this with. It's not too bad.

Here is a general case, then you can use whatever V you wish.

Assuming it has a top and bottom:

$\displaystyle S=2{\pi}r^{2}+2{\pi}rh$........[1]

$\displaystyle V={\pi}r^{2}h$.......[2]

Solve [2] for h and sub into [1]:

$\displaystyle h=\frac{V}{{\pi}r^{2}}$........[3]

$\displaystyle S=2{\pi}r^{2}+2{\pi}r(\frac{V}{{\pi}r^{2}})$

$\displaystyle =2{\pi}r^{2}+\frac{2V}{r}$

$\displaystyle \frac{dS}{dr}=4{\pi}r-\frac{2V}{r^{2}}$

Set this to 0 and solve for r and we get $\displaystyle r=\frac{2^{\frac{2}{3}}V^{\frac{1}{3}}}{2{\pi}^{\f rac{1}{3}}}$

Sub into [3] and find $\displaystyle h=\frac{2^{\frac{2}{3}}V^{\frac{1}{3}}}{{\pi}^{\fr ac{1}{3}}}$

See?. h is twice r just as was stated in the beginning.

3. ## Thanks

Thanks Galactus, I'll give it a try and hopefully it works. =)