help with applied maximum and minimum problem?

[christinax77]

Find the dimensions of the closed cylindrical can that will have a capacity of k units³ and will use the minimum amount of material. Find the ratio of the height h to the radies r of the top and bottom.

I'm confused with this problem because it does not give any values to substitute for the variables.

 

[skeeter]

\(\displaystyle V = \pi r^2 h\)

\(\displaystyle k = \pi r^2 h\)

\(\displaystyle h = \frac{k}{\pi r^2}\)

\(\displaystyle A = 2\pi r^2 + 2\pi r h\)

substitute \(\displaystyle \left(\frac{k}{\pi r^2}\right)\) for h in the surface area formula ...

\(\displaystyle A = 2\pi r^2 + 2\pi r \frac{k}{\pi r^2}\)

\(\displaystyle A = 2\pi r^2 + \frac{2k}{r}\)

find \(\displaystyle \frac{dA}{dr}\) and determine the value of r that minimizes A ... remember to treat k as a constant.