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Storing Sequestrates

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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]

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