Cylinder with its rims on a sphere: surface area with double-angle identities, an angle from a tangent, and the exact volume

A solid cylinder has base radius \(R\) cm and height \(h\) cm. It is placed inside a hollow sphere with centre \(O\) and radius \(r\) cm so that the circular edge of each end of the cylinder touches the inner surface of the sphere, and the axis of the cylinder passes through \(O\).

The point \(M\) is the centre of the top face of the cylinder, and \(A\) is a point on the edge of that face, so that \(A\hat{O}M=\theta\), where \(0<\theta<\frac{\pi}{2}\). This arrangement is shown in the diagram below.

1. 1. Using triangle \(OMA\), show that \(R=r\sin\theta\).
   2. Find \(h\) in terms of \(r\) and \(\theta\).[4]
2. Hence or otherwise, show that the total surface area of the cylinder, \(S\text{ cm}^2\), satisfies \(S=\pi r^2(1-\cos2\theta+2\sin2\theta)\).[4]

It is now given that \(S=\frac75\pi r^2\).

3. Show that \(\tan\theta=\frac13\).[4]

Let the volume of the cylinder be \(V\text{ cm}^3\).

4. Find \(V\) in the form \(p\pi r^3\sqrt{10}\), where \(p\in\mathbb{Q}^+\).[5]