A hollow container is to be made out of a fixed total area \(\pi a^2\) of sheet metal and its shape is to be that of a right circular cone completed by the circular base on which it stands. Find the radius of the base when the container encloses maximum volume.

Drawing a diagram will probably help.

Can we write down or work out a formula for the surface area of a cone?

Can we derive formulae for the height and volume of the cone from this?