A closed cylinder has a circular base of radius $r.$ The height of the cylinder is $h.$

Given that the surface area of the cylinder is $60{\text{cm}}^2,$ find the maximum volume of the cylinder.

If you open out the cylinder, you see it is made from two circles and a rectangle, whose width is the circumference of the circular top.

The solution can be given as

for square-free $a\in \mathbb{N}$. Find the value of $a$.