2025 CGS PRELIMS P1 Q9

The diagram shows a container, made of a thin sheet of plastic, in the shape of an open half-cylinder, with no rectangular lid. The length of the container is \(y\) cm and each of the semicircular ends has radius \(r\) cm. The total external area of the plastic used is \(8500\text{ cm}^2\).

1. Show that the volume, \(V\text{ cm}^3\), of the container is given by \(4250r - \frac{1}{2}\pi r^3\).[3]
2. Given that \(r\) can vary, find the stationary value of \(V\) and determine its nature.[4]