(A Math) 2025 CCHY PRELIMS P1 Q7

The diagram shows a solid cylinder of radius \(r\) cm and height \(h\) cm inscribed in a hollow cone of height 45 cm and base radius 20 cm. The cylinder rests on the base of the cone and the circumference of the top surface of the cylinder touches the curved surface of the cone.

1. Show that the volume, \(V\text{ cm}^3\), of the cylinder is given by \(V=45\pi r^2-\dfrac{9}{4}\pi r^3\).[3]
2. Given that \(r\) can vary, find the maximum volume of the cylinder, leaving your answer in terms of \(\pi\).[5]
3. Hence show that the cylinder occupies at most \(\dfrac{4}{9}\) of the volume of the cone.[2]