A manufacturer produces a tank for storing liquid CO_{2} underground.

The tank is modelled in the shape of a hollow vertical circular cylinder closed with a flat lid at the top and a hemispherical shell at the bottom.

The walls of the tank are assumed to have negligible thickness.

The cylinder has a radius *r* metres and height *h* metres and the hemisphere has radius *r* metres.

The volume of the tank is 6 m^{3}.

a) Show that, according to the model, the surface area of the tank, in m^{2} is given by

\[\frac{12}{r} + \frac{5}{3}\pi r^{2}\]

[4 marks]

The manufacturer needs to minimise the surface area of the tank, to minimise costs.

b) Find, using calculus, the radius of the tank for which the surface area is a minimum.

[4 marks]

c) Find the surface area of the tank for this radius, giving your answer to the nearest integer.

[2 marks]