Calculus

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $60/ft and on the other three sides by a metal fence costing $50/ft. If the area of the garden is 102 square feet, find the dimensions of the garden that minimize the cost.

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  1. let the width be x
    and the length y , were the brick wall is along the length.

    the xy = 102
    y = 102/x

    Cost = 50(2x) + 50y + 60y
    = 100x + 110y
    = 100x + 110(102/x) = 100x + 11220/x
    d(cost)/dx = 100 - 11220/x^2
    = 0 for a min of cost
    11220/x^2 = 100
    100x^2 = 11220
    x^2 = 11220/100
    x = 105.925/10

    x = 10.59 ft
    they y = 102/x = 9.63 ft

    The math tells me that the wall should be build along the shorter side (makes sense)

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