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A fence 5 feet tall runs parallel to a tall building at a distance of 7 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? [Hint: Determine the length of a ladder that touches the building, fence, and ground as a function of the acute angle the ladder makes with the ground.]

  • Calculus -

    make a sketch showing the ladder touching the fence and making contact with the wall.
    let the angle at the base of the ladder be ß
    draw a horizontal from the top of the fence to the wall, then the angle the ladder makes with that line is ß, giving up 2 similar right-angled triangles

    the ladder has length L1 above the fence and L2 below the fence
    cosß = 7/L1 ---> L1 = 7secß
    sinß = 5/L2 ---> L2 = 5cscß
    so L = 7secß + 5sinß = 7(cosß)^-1 + 5(sinß)^-1

    dL/dß = -7(cosß)^-2(-sinß) - 5(sinß)^-2(cosß)
    simplifying and setting equal to zero gives me

    7sinß/(cosß)^2 = 5cosß/(sinß)^2

    7(sinß)^3 = 5(cosß)^3 or
    (tanß)^3 = 5/7
    tanß = (5/7)^(1/3) = .8939..
    ß=41.7936 degrees

    sub that back into L1 and L2, add them up , ....

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