A uniform solid consists of a hemisphere with centre $O$ and radius $0.6{\text{ }}m$ joined to a cylinder of radius $0.6{\text{ }}m$ and height $0.6{\text{ }}m$. The plane face of the hemisphere coincides with one of the plane faces of the cylinder.

a) Calculate the distance of the centre of mass of the solid from $O$.

[The volume of a hemisphere of radius $r$ is $\frac{2}{3}\pi {r^3}$]

b) A cylindrical hole, of length $0.48{\text{ }}m$, starting at the plane face of the solid, is made along the axis of symmetry (see diagram). The resulting solid has its centre of mass at $O$. Show that the area of the cross-section of the hole is $\frac{3}{{16}}\pi {m^2}$.

c) It is possible to increase the length of the cylindrical hole so that the solid still has its centre of mass at $O$. State the increase in the length of the hole.