8 A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is \(192 \pi \mathrm {~cm} ^ { 2 }\). The cylinder has a radius of \(r \mathrm {~cm}\) and a height of \(h \mathrm {~cm}\).
- Express \(h\) in terms of \(r\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by
$$V = \frac { 1 } { 2 } \pi \left( 192 r - r ^ { 3 } \right) .$$
Given that \(r\) can vary,
- find the value of \(r\) for which \(V\) has a stationary value,
- find this stationary value and determine whether it is a maximum or a minimum.