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Getting stuck on a certain part of this question:

Q. A manufacturer wants to produce one litre (1000ml) cans from as small a quantity of metal sheeting as possible to reduce costs.

What I've done so far:

I've shown that the height of the cylinder is:

$h=\displaystyle\frac{1000}{\pi r^2}$

I've shown that the total surface area is:

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

Derivative:

$A' = 4\pi r - 2000r^{-2}$

Then found the zero or the x coordinate of the minimum value:

$r=\left(\displaystyle\frac{500}{\pi}\right)^{\frac{1}{3}}$

I am now trying to sub

$r=\left(\displaystyle\frac{500}{\pi}\right)^{\frac{1}{3}}$ into $A = 2\pi r^2+\displaystyle\frac{2000}{r} $ to find the minimum surface area of the 1000ml can. I'm just getting stuck with the algebra.

The answer I get is $A = 3000\left(\displaystyle\frac{500}{\pi}\right)^{\frac{-1}{3}}$ which is correct because I graphed it using a graphing calculator and both give $553.581$

The answer in the back of the book gives $A=6\pi\left(\displaystyle\frac{500}{\pi}\right)^{\frac{2}{3}} $

Can someone please show me the steps to get the answer in the back of the book when subbing $r$ into $A$

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1 Answer

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Your answers are both correct and identical. Since $-1/3=2/3-1$: $$A= 3000 \left(\frac{500}{\pi}\right)^{-1/3} = 3000 \left(\frac{500}{\pi}\right)^{-1}\left(\frac{500}{\pi}\right)^{2/3} = 3000 \left(\frac{\pi}{500}\right)\left(\frac{500}{\pi}\right)^{2/3} $$ $$=6\pi\left(\frac{500}{\pi}\right)^{2/3}$$

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