A cylindrical tank is to have a capacity of 1000 m³. It is to fit into a foundry that is 12 m wide with a height of 11 m.

The base of the tank will cost half as much as the top. The metal for the side of the tank will cost four fifths as much as the top.

An equation that can be used to model the cost of the tank is C(r) = (3/2) pi(r)^2 + 1600/r. Determine what radius value will give a minimum cost. You must prove this value gives a minimum.

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  1. well, we know that thee will be a min or max when dC/dr = 0
    C' = 3πr - 1600/r^2 = (3πr^3-1600)/r^2
    C'=0 when r = ∛(1600/(3π)) ≈ 5.537
    Now just check the sign of C" to see whether it's a min or a max

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