Cylindrical can of minimum surface

Text

A cylindrical can must hold $1$ litre $=1000\ \text{cm}^3$. Find the radius and height minimizing the total surface (including the two bases).

Solution

Volume: $\pi r^{2}h=1000\Rightarrow h=\dfrac{1000}{\pi r^{2}}$. Surface: $S(r)=2\pi r^{2}+\dfrac{2000}{r}$. $S'(r)=4\pi r-\dfrac{2000}{r^{2}}=0\Rightarrow r^{3}=\dfrac{500}{\pi}\Rightarrow r=\sqrt[3]{\tfrac{500}{\pi}}\approx5.42\text{ cm}$. Then $h\approx10.84\text{ cm}$ (note $h=2r$). $\boxed{r\approx5.42\ \text{cm};\quad h\approx10.84\ \text{cm}}$