2025 May TZ1 P1 Q9

A solid cylinder has height \(h\) cm and base radius \(R\) cm.
The cylinder fits exactly inside a hollow sphere of radius \(r\) cm.
Points \(\mathrm{A, B}\) and \(\mathrm{C}\) are points where the surface of the cylinder touches the surface of the sphere.
The line segment \([\mathrm{AB}]\) is a diameter of the sphere.
The line segment \([\mathrm{BC}]\) is a diameter of the base cylinder and \(\mathrm{A\hat{B}C} = \theta\).
This information is shown on the following diagram.

1. [4]

   1. By considering triangle \(\mathrm{ABC}\), show that \(R = r\cos{\theta}\).

   2. Find an expression for \(h\) in terms of \(r\) and \(\theta\).

2. Hence or otherwise, show that the total surface area, \(S\) cm\(^2\), of the cylinder is given by \(S = 2\pi r^2(1 + 2\sin{\theta}\cos{\theta} - \sin^2{\theta})\).

   [4]

The external surface area of the sphere is \(2S\).

3. Show that \(\tan{\theta} = 2\).

   [4]

The volume of the cylinder is \(V\) cm\(^3\).

4. Find \(V\), giving your answer in the form \(p \pi r^3 \sqrt{5}\), where \(p \in \mathbb{Q}^+\).

   [5]