2022 SAJC Promo Q8

[Volume of a cylinder, \(V=\pi {{r}^{2}}h\), where \(r\) is the radius of the cylinder and \(h\) is the vertical height]

A manufacturer wants to produce a container in a shape of a closed cylinder of radius \(r\) cm, height \(h\) cm with a fixed volume \(400\)cm\(^{3}\).

1. Show that its surface area, \(S\), is given by \(2\pi {{r}^{2}}+\frac{800}{r}\).[2]
2. Use differentiation to find the exact minimum value of \(S\), proving that it is a minimum. Give your answer in the form \(a{{\left( b \right)}^{\frac{2}{3}}}{{\left( \pi \right)}^{\frac{1}{3}}}\), where \(a\) and \(b\) are positive integers. Find also the ratio \(r:h\) that gives this minimum value of \(S\).[8]