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Math Help - Need help in optimization

  1. #1
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    Need help in optimization

    1. A ladder simultaneously leans against a 4.0m wall and a tall building that is 1.0 m behind the wall. Find the length of the shortest ladder that can be used. How high against the building will the leader reach in this case?

    Do the optimization in terms of the angle that the ladder makes with the ground.

    Diagram:



    So basically I am asked to express both L1 + L2 in terms of theta where L = L1 + L2 and theta = u

    Using similar triangles:
    4/x = h/x+1

    and:

    L1 = 4/sin u
    L = h/sin u
    L2 = h/sin u - L1

    If I am going about this correctly what should I do next? I mean I'm at a complete loss here as there are just too many unknowns and I know that in optimization you should reduce the variable to just one which would be theta. How would I go about in doing this?
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  2. #2
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    using pythagoras' theorum

    L_1 = \sqrt{x^2 - 4^2}



    Using similar Triangles
    L = L_1 * \frac{1+x}{x}



    combine equations
    L = \frac{(1+x)\sqrt{x^2 - 4^2}}{x}

    Can you differenciate that (may want to simplify first)?
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  3. #3
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    I need a function in terms of theta though and im not sure if the one i gave is correct. Also L1 = sqrt of x^2 + 4^2.
    Last edited by GameTheory; July 21st 2010 at 02:01 PM.
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  4. #4
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    You have an expression in x (if u fix the pythag as you suggested)

    You can rewrite this as a function in theta if you note that
    X= \frac{4}{tan(\theta)}

    i wouldn't like to differenciate that though!
    Last edited by SpringFan25; July 22nd 2010 at 02:44 AM. Reason: removed links to similar problems. solution now below
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  5. #5
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    here is an easier way, i modified your diagram a bit:

    Need help in optimization-math-2.png



    From this, you can see that the length of the ladder is:

    L = \frac{4}{sin(\theta)} + \frac{1}{cos(\theta)}

    You can differenciate that with respect to theta.
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