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Math Help - Optimization Problem

  1. #1
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    Optimization Problem

    Find the maximum horizontal overhang for a 15 foot ladder over an 8 foot fence

    The ladder base is always on the ground, the fence is vertical and ground it horizontal
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  2. #2
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    Quote Originally Posted by Exiab View Post
    Find the maximum horizontal overhang for a 15 foot ladder over an 8 foot fence

    The ladder base is always on the ground, the fence is vertical and ground it horizontal
    1. Draw a sketch. Let x denote the base of the ladder and y the overhang.

    2. Use proportions:

    \frac8x = \frac z{x+y}~\implies~z=\frac{8(x+y)}{x}

    3. Use Pythagorean theorem:

    (x+y)^2 + z^2=15^2~\implies~(x+y)^2+\left(\frac{8(x+y)}{x}\r  ight)^2=225

    4. Solve for y. That means: Calculate the overhang with respect to the base. You'll get a positive and a negative result. The negative result isn't very plausible here. Thus:

    y = \frac{15x}{\sqrt{x^2+64}}-x

    5. Calculate the first derivative (it's easier (for me!) to use product rule and chain rule):

    y' = \frac{960}{\left(\sqrt{x^2+64}\right)^3}-1

    6. Solve for x: y' = 0. I've got x = 4 \cdot \sqrt{\sqrt[3]{225}-4} \approx 5.77193...
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  3. #3
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    Let F denote the foot of the ladder which is moved horizontally.

    Then the top of the ladder describes a curve which I have "recorded". The vertical tangent to this curve indicates the maximum overhang.

    According to my previous calculations the overhang is y = 3.004583...
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  4. #4
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    Quote Originally Posted by earboth View Post
    Let F denote the foot of the ladder which is moved horizontally.

    Then the top of the ladder describes a curve which I have "recorded". The vertical tangent to this curve indicates the maximum overhang.

    According to my previous calculations the overhang is y = 3.004583...
    This answer is consistent with that found using the method suggested by Opalg at http://www.mathhelpforum.com/math-he...imization.html.
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    This answer is consistent with that found using the method suggested by Opalg at http://www.mathhelpforum.com/math-he...imization.html.
    Opalg's method is much more elegant than mine. Unfortunately such solutions never occur to me ...
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  6. #6
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    Quote Originally Posted by earboth View Post
    Opalg's method is much more elegant than mine. Unfortunately such solutions never occur to me ...
    Well, on the other hand:

    1. You always have lovely graphics (it really is your trademark)

    2. You reply is dated 11 March 11:20 am and Oplag's is dated 14 March 12:32 am. So it took Opalg 3 more days than you to get the elegant solution Lives might have been lost if not for your less elegant but timely solution


    (ps Only kidding, opalg )
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