2015 AJC H1 Q3

To store a special liquid chemical, a manufacturer constructs a closed cylindrical container of negligible thickness with base radius r cm and an internal surface area of \(96\pi \) cm\(^{2}\).

1. Show that the volume of liquid chemical the container can store as \(r\) varies, is given by \(V=\pi r\left( 48-{{r}^{2}} \right)\,\,\mathrm{c}{{\mathrm{m}}^{3}}\).[1]
2. Using a non-calculator method, find the maximum value of \(V\).[3]

The manufacturer decides to construct the closed cylindrical container with a fixed radius of \(k\) cm. When liquid chemical is poured into the container at a constant rate of \(2\) cm\(^{3}\)s\(^{-1}\), the height of the liquid chemical in the container increases at a constant rate of \(0.5\) cms\(^{-1}\). Find \(k\) in terms of \(\pi \mathrm{.}\)[3]