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Math Help - application ladder on wall

  1. #1
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    application ladder on wall

    problem:
    A ladder is to reach over a fence 27/8 meters high to a wall 8 meters behind the fence. Find the length of the shortest ladder that may be used.
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  2. #2
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    Quote Originally Posted by cazimi View Post
    problem:
    A ladder is to reach over a fence 27/8 meters high to a wall 8 meters behind the fence. Find the length of the shortest ladder that may be used.

    Hello,

    I've modified your sketch: I added a few names.

    By similar triangles you get the proportion:
    27/8.....y
    ----- = ---
    ... x.......x+8

    and you'll get: y = 27/8 + 27/x

    The length of the ladder can be calculated by (use Pythagoran theorem):

    l = (x+8) + (27/8 + 27/x)

    l(x) = x + 16x + 64 + 729/64 + 729/(4x) + 729/(x)

    If you want to know the shortest ladder possible you have to calculate a minimum. Therefore you calculate the derivative of l(x) which has to be equal to zero: (If l has a minimum, that means l'(x) = 0 then l has a maximum, that means (l)'(x) = 0)

    (l)'(x) = 2x + 16 - 729/(4x) - 729/(x). Now (l)'(x) = 0:

    2x + 16 - 729/(4x) - 729/(x) = 0. Multiply by x

    2x^4 + 16x - 729/4 x - 1458 = 0

    2x(x + 8) - 729/4(x+8) = 0

    (2x - 729/8)(x + 8) = 0

    x = -8 this result is not very plausible with your problem

    x = √(729/8) = 9/2

    Plug in this value into the equation for l:

    l = (9/2 + 8) + (27/8 + 27/(9/2))

    l = 156.25 + 5625/64 = 15625/64

    Therefore the ladder has the length: l = √(15625/64) = 125/8m ≈ 15.63 m

    EB



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