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A fence 2 feet tall runs parallel to a tall building at a distance of 6 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

  • Calculus -

    Let the ladder length be d
    Let the base of the ladder be x feet from the fence, and hit the wall at height h.

    from similar triangles,

    2/x = h/(6+x)
    h = 2(x+6)/x

    d^2 = (6+x)^2 + h^2
    = (x+6)^2 + 4(x+6)^2/x^2
    = (x+6)^2 * (1 + 4/x^2)

    d^2 (and therefore d) has a minimum at x = 2.9

    d^2 = 116.88
    d = 10.8 ft

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