2021 ONE(e)

A cone has a height of 3 m and a radius of 1.5 m.

A cylinder is inscribed in the cone, as shown in the diagram below.

The base of the cylinder has the same centre as the base of the cone.

Prove that the maximum volume of the cylinder is $\pi$ m${}^3$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*