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A rancher wants to fence in an area of 5189400 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?

  • Calculus -

    Let x and y be the two side lengths of the full rectangle. The length x will be divided in two. The total lengths of fence needed will be

    L = 2x + 3y.

    You know that x y = 5,189,400 ft^2

    dL/dx = d/dx (2x + 3*5,189,400/x)
    = 2 - 15,568,200/x^2 = 0 at minimum L
    x^2 = 7,784,100
    x = 2790 ft y = 1860
    Quantity of fence needed = 2x + 3y = ?

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