4 Machines in a factory make cardboard cones of base radius \(r \mathrm {~cm}\) and vertical height \(h \mathrm {~cm}\). The volume, \(V \mathrm {~cm} ^ { 3 }\), of such a cone is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\). The machines produce cones for which \(h + r = 18\).
- Show that \(V = 6 \pi r ^ { 2 } - \frac { 1 } { 3 } \pi r ^ { 3 }\).
- Given that \(r\) can vary, find the non-zero value of \(r\) for which \(V\) has a stationary value and show that the stationary value is a maximum.
- Find the maximum volume of a cone that can be made by these machines.