Hi,

The problem is:

given a cylinder (I prefer beer can) holding 1000 cubic cm. What should the radius and height be to minimise the surface area.

I have tried the following

$\displaystyle V=\pi r^2h, V=1000$

therefore,

$\displaystyle h=\frac{1000}{\pi r^2}$ and $\displaystyle S=2\pi r^2+2\pi rh$

subsitituting (h), and simplifying

$\displaystyle S=2 \pi r^2+\frac{2000}{r}$

derive that to get:

$\displaystyle S'=4\pi r-\frac{2000}{r^2}$

Solving for r when S'=0 (critical point), I obtain.

$\displaystyle r=\sqrt[3]{\frac{500}{\pi}}$, which according to the text book is correct. Great news. Now I get stuck...

I cannot get h correct. For some reason h is $\displaystyle 2\sqrt[3]{\frac{500}{\pi}}$.

I am so close yet cannot figure out how to get that result.

Subsituting r into the original equation for h gives:

$\displaystyle \frac{1000}{\pi [\sqrt[3]{\frac{500}{\pi}}]^2}$

This is not correct.

Thanks

Craig.