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March 28, 2017

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a fence 8 ft tall stands on level ground and runs parallel to a tall building. if the fence is 1 ft from the building, find the length of the shortest ladder that will extend from the ground over the fence to the wall of the building. (hint #1: if L represents the length of the ladder, the quantity L is minimized when the quantity L2 is minimized, so you do not have to concern yourself with thte square root-just minimize L2 and that will minimize L. Hint#2: use similar triangles)
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  • calculus - ,

    Make a diagram
    let the foot of the ladder be x ft from the fence
    let the ladder reach y ft above the ground

    I see similar triangle so set up a ratio
    8/x = y/(x+1)
    xy = 8x+8
    y = (8x+8)/x

    let the length of the ladder be L
    L^2 = (x+1)^2 + y^2
    = (x+1)^2 + [(8x+8)/x]^2
    = x^2+2x+1 + 64 + 128/x + 64/x^2

    2L dL/dx = 2x + 2 - 128/x^2 - 128/x^3
    = 0 for a min of L

    2x + 2 - 128/x^2 - 128/x^3 = 0
    multiply by x^3
    2x^4 + 2x^3 - 128x - 128 = 0
    2x^3(x+1) - 128(x+1) = 0

    (x+1)(2x^3 - 128) = 0
    x = -1 , not likely
    or
    2x^3=128
    x^3=64
    x=4

    sub into L^2
    L^2 = 5^2 + 10^2 = 125

    I minimized L^2

  • calculus - ,

    12.727 ft

  • calculus - ,

    Make a diagram
    let the foot of the ladder be x ft from the fence
    let the ladder reach y ft above the ground

    I see similar triangle so set up a ratio
    8/x = y/(x+1)
    xy = 8x+8
    y = (8x+8)/x

    let the length of the ladder be L
    L^2 = (x+1)^2 + y^2
    = (x+1)^2 + [(8x+8)/x]^2
    = x^2+2x+1 + 64 + 128/x + 64/x^2

    2L dL/dx = 2x + 2 - 128/x^2 - 128/x^3
    = 0 for a min of L

    2x + 2 - 128/x^2 - 128/x^3 = 0
    multiply by x^3
    2x^4 + 2x^3 - 128x - 128 = 0
    2x^3(x+1) - 128(x+1) = 0

    (x+1)(2x^3 - 128) = 0
    x = -1 , not likely
    or
    2x^3=128
    x^3=64
    x=4

    sub into L^2
    L^2 = 5^2 + 10^2 = 125

    I minimized L^2

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