Differentiation - Problems Involving Maximisation and Minimisation

The diagram shows a right circular cone within a sphere of centre \(O\) and radius \(8\) cm.

The vertex of the cone, \(A\), is on the sphere and the centre of the sphere is within the cone.

The radius of the base of the cone is \(r\) cm.

1. If the distance of \(O\) from the base of the cone is \(h\) cm, express \(r\) in terms of \(h\).
2. Given that the volume of the cone is \(V\) cm\(^3\), show that \(V = \frac{\pi}{3} \left( 8 + h \right)^2 \left( 8 - h \right)\).
3. Given that \(r\) and \(h\) can vary,
   1. find the stationary value of \(V\), leaving your answer to the nearest cm\(^3\).
   2. determine whether this stationary value is a maximum or a minimum.