calculus

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A fence 8 feet tall stands on level ground and runs parallel to a tall building. If the fence is 1 foot from the building, find the shortest ladder that will extend from the ground over the fence to the wall of the building.

  • calculus -

    I assume you made a sketch.
    let the distance from the fence to the foot of the ladder be x
    let the height of the ladder up the wall be h
    by ratios:
    8/x = h/(1+x)
    hx = 8(1+x
    h = (8+8x)/x

    let the length of the ladder be L
    L^2 = h^2 + (1+x)^2
    = (8+8x)^2/x^2 + (1+x)^2
    = (64 + 128x + 64x^2)/x^2 + 1 + 2x + x^2
    = 64/x^2 + 128/x + 64 + 1 + 2x + x^2
    2L dL/dx = -128/x^3 - 128/x^2 + 2 + 2x
    = 0 for a min of L
    times -x^3
    128 + 128x - 2x^3 - 2x^4 = 0
    128(1+x) - 2x^3(1+x) = 0
    (1+x)(2x^3 - 128) = 0
    x = -1 , which makes no sense
    or
    2x^3 - 128 = 0
    x^3 = 64
    x = 4

    So L^2 = (8 + 32)^2 /4 + (1+4)^2
    = 400 + 25
    L^2 = 425
    L = √425 = 5√17 = appr 20.6 ft

    check my arithmetic.

  • calculus -

    This solution is correct but isn't the last step wrong because isn't it (8+32)^2/4^2 instead of (8+32)^4/4?

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