Differentiation - Problems Involving Maximisation and Minimisation

The diagram shows the cross-section of a conical vessel in the shape of an inverted right circular cone \(OAB\) of height \(60\) cm and base radius \(18\) cm. A cylindrical container \(PQRS\) of radius \(r\) cm and height \(h\) cm is fitted into the cone such that the upper edge of the cylinder is in contact with the vessel.

1. Show that the volume, \(V\) cm\(^3\), of the cylinder is given by \(V = \pi \left( 60r^2 - \frac{10}{3}r^3 \right)\).
2. Given that \(r\) can vary, find the radius of the largest cylinder which can fit into the vessel and hence show that the cylinder occupies \(\frac{4}{9}\) of the volume of the cone.