2008 AJC P1 Q7

A capsule made of two identical hemispheres of radius \(r\) cm, one at each end of a cylinder, is inscribed into a fixed cone of height \(12\) cm and base radius \(9\) cm. The capsule and the cone have a common axis of symmetry, and the cross-section is as shown.

1. Show that the height, \(x\) cm, of the cylinder of the capsule, is related to the equation \(x=12-\frac{8}{3}r\).
2. Given that when \(r=2\)cm, \(x\) is decreasing at a rate of \(0.1\)cm/s, find the rate of change at this instant of the volume of the capsule.