A manufacturer produces a tank for storing liquid CO2 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 m3.
a) Show that, according to the model, the surface area of the tank, in m2 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]
