# Thread: Finding the height of a minimised cylinder.

1. ## Finding the height of a minimised cylinder.

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

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

subsitituting (h), and simplifying
$S=2 \pi r^2+\frac{2000}{r}$

derive that to get:
$S'=4\pi r-\frac{2000}{r^2}$

Solving for r when S'=0 (critical point), I obtain.
$r=\sqrt[3]{\frac{500}{\pi}}$, which according to the text book is correct. Great news. Now I get stuck...

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

I am so close yet cannot figure out how to get that result.
Subsituting r into the original equation for h gives:

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

This is not correct.

Thanks
Craig.

2. Hello,

$h=\frac{1000}{\pi \left(\sqrt[3]{\frac{500}{\pi}}\right)^2}$

Now let's do some bits of algebra... remember this property :
$\sqrt[n]{x}=x^{1/n}$
and this : $(a^b)^c=a^{bc}$

So $h=\frac{2 \cdot 500}{\pi \left(\frac{500}{\pi}\right)^{2/3}}$

$h=\frac{2 \cdot 500}{\pi \cdot 500^{2/3} \cdot \pi^{-2/3}}$

$h=2 \cdot \frac{500 \cdot 500^{-2/3}}{\pi \cdot \pi^{-2/3}}$

$h=2 \cdot \frac{500^{1/3}}{\pi^{1/3}}$

$h=2 \sqrt[3]{\frac{500}{\pi}}$

3. ## Lessons,

Hi,

Thanks, these are the school fees. I must learn to persevere.

Cheers...