3.70: Cone volume maximisation with fixed curved surface

The volume, $V$, of a cone with base radius $r$ and vertical height $h$ is given by $\dfrac{1}{3}\pi r^{2} h$. The curved surface area of a cone with base radius $r$ and slant height $l$ is given by $\pi r l$.

A cone has base radius $r\,\mathrm{cm}$, vertical height $h\,\mathrm{cm}$ and volume $V\,\mathrm{cm}^{3}$. The curved surface area of the cone is $4\pi\,\mathrm{cm}^{2}$.