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Thread: Decide shortest length of ladder

  1. #1
    Senior Member Twig's Avatar
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    Decide shortest length of ladder

    Hi

    I have attached a picture which help describe the problem.

    The problem is: A fence is located 3m from a wall, and the fence is 2m high.
    Decide the shortest length of a ladder (in green in the picture) placed on the ground and over the fence, touching the wall.



    There are a lot of congruent right triangles, will this be used in the solution?
    If I name the distance from the fence to the position where the ladder touches the ground a, and the height where the ladder touches the wall for $\displaystyle (h+2) $ . (The $\displaystyle + 2 $ comes from the fence height).

    Wonīt the length of the ladder be given by the pythagoran theorem?

    Like

    $\displaystyle \sqrt{(a+3)^{2}+(h+2)^{2}} $

    Thx
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  2. #2
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    Absolutely! And so use your congruent triangles idea to get either one of a or h as a function of the other. Then you can see how the ladder length varies as a function of a or h...


    Ouch! (below...)
    Quite right
    Last edited by tom@ballooncalculus; Jul 4th 2009 at 10:53 AM.
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  3. #3
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    Not "congruent" triangles- similar triangles. In particular, $\displaystyle \frac{a}{2}= \frac{h+3}{a+3}$
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  4. #4
    Senior Member Twig's Avatar
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    Hi

    Yes, similar , thats what I meant =) Just didnīt come up with the english word

    I think I got it now, but the answer becomes really messy..
    I get a minima for $\displaystyle a=\sqrt[3]{12}$

    Now I have $\displaystyle h = 2\cdot \frac{(a+3)}{a}-2 $

    So I plug these into $\displaystyle \sqrt{(a+3)^{2}+(h+2)^{2}} $ and ugh, I donīt care to write it out, itīs not important. If the solution is correct then I am satisfied.

    Thx
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  5. #5
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    Surely,

    $\displaystyle \frac{h + 2}{a + 3} = \frac{2}{a}$

    giving

    $\displaystyle h = \frac{6}{a}$

    ?
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  6. #6
    Senior Member Twig's Avatar
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    Hi Tom

    Yes, that is also correct.

    Differentiating with $\displaystyle h=\frac{6}{a} $ gives $\displaystyle a=\sqrt[3]{12} $

    And here it is obv easier to obatin a value for $\displaystyle h $
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  7. #7
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    Well done.
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