Differentiation - Connected Rates of Change

A capsule comprises of two identical hemispheres and a cylinder of radius \(r\) cm, which is inscribed in a cone of fixed height \(12\) cm and fixed radius \(9\) cm. Assume that the capsule and the cone have a common axis of symmetry. The cross-section is shown in the diagram.

1. Let \(x\) cm be the height of the cylindrical part of the capsule. Show that \(3x + 8r = 36\).
2. Given that when \(r = 2\) cm, \(x\) is increasing at a rate of \(0.1\) cm/s, find the rate of change of the volume of the capsule at this instant.