Problem

A manufacturer wishes to design an open cylindrical tank, as shown in the diagram. The tank will have a base but no top. The outside of the tank will have a fixed surface area of $600\pi\text{ cm}^2$. The radius $r$ cm and height $h$ cm of the tank can vary.

(a) Show that the volume, $V\text{ cm}^3$, of the tank is given by

$$V=\frac{\pi r(600-r^2)}{2}.$$

(b) Find the exact value of $r$ which corresponds to the maximum value of $V$.

(c) Hence, find the maximum value of $V$.