A sphere of radius \(\quantity{6}{cm}\) fits inside a cone of vertical angle \(84^\circ\), so that \(BA\) is a tangent to the sphere.
![The sphere and cone described in the question; the point $B$ is the apex of the cone and $A$ lies on the intersection between the sphere and the base circle of the cone.](/thinking-about-geometry/r9694/images/diagram-labelled.png)
Calculate
- the length \(AB\);
- the radius of the base of the cone;
- the shortest distance from \(B\) to the sphere;
- the volume of the cone.