Calculus/Optimisation

An open water tank with a square base is to be made from a thin sheet of metal.
find the length of the square base and the height of the tank so that the least amount of metal is used to make a tank of capacity 8 meters cubed

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  1. let the base be x m by x m , and let the height be y m
    V= x^2y
    8 = x^2y ---> y = 8/x^2

    Surface area
    = SA = x^2 + 4xy
    = x^2 + 4x(8/x^2)
    = x^2 + 32/x
    d(SA)/dx = 2x - 32/x^2
    = 0 for a min of SA
    2x = 32/x^2
    x^3 = 16
    x = 16^(1/3) = appr 2.52 m
    y = .....

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