9

9. [Maximum mark: 17]
   Consider a cylinder of radius $4r$ and height $h$. A smaller cylinder of radius $r$ is removed from the centre to form a hollow cylinder. This is shown in the following diagram.

All lengths are measured in centimetres.

The total surface area of the hollow cylinder, in cm², is given by $S$.
The volume of the hollow cylinder, in cm³, is given by $V$.

(a) Show that $S = 30\pi r^{2} + 10\pi r h$. [3]

(b) The total surface area of the hollow cylinder is $240\pi$ cm².
Show that $V = 360\pi r - 45\pi r^{3}$. [6]

(c) Find an expression for $\dfrac{dV}{dr}$. [2]

The hollow cylinder has its maximum volume when $r = p\sqrt{\dfrac{2}{3}}$, where $p \in \mathbb{Z}^{+}$.

(d) Find the value of $p$. [3]

(e) Hence, find this maximum volume, giving your answer in the form $q\pi\sqrt{\dfrac{2}{3}}$, where $q \in \mathbb{Z}^{+}$. [3]