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A fence 5 feet tall runs parallel to a tall building at a distance of 3 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 -

    draw a diagram. If the foot of the ladder is x feet from the fence, and the ladder reaches y feet up the wall, then we have

    x/5 = (x+3)/y
    y = 5 + 15/x

    if the length of the ladder is z,

    z^2 = (x+3)^2 + y^2
    = (x+3)^2 + (5 + 15/x)^2
    z = √((x+3)^2 + (5 + 15/x)^2)

    dz/dx = (x+3)(x^3-75) / <a bunch of nonzero junk>

    so, dz/dx=0 at x = -3 or ∛75

    z^2 = (3+∛75)^2 + (5+15/∛75)^2
    z = 11.19 feet

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