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Math Help - Maximization

  1. #1
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    Maximization

    How far from A should the point P be chosed so as to maximize the angle theta?



    I've been looking at this for awhile and don't know how to begin. Any pointers would be greatly appreciated. Thanks.
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  2. #2
    MHF Contributor
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    The angles around theta add up to 180 (they are on a straight line)

    180=\theta + \arctan{\frac{5}{AP}} + \arctan{\frac{2}{3-BP}}

    \theta = 180 - \arctan{\frac{5}{AP}} -  \arctan{\frac{2}{3-AP}}

    Can you differenciate that? looks like it might get messy...
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  3. #3
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    use a^2 + b^2 - c^2 = 2abcos\theta with a the length of the hypotenuse defined by \sqrt{(3-P)^2 + 2^2}; b the length of the hypotenuse defined by \sqrt{P^2 + 5^2}; and c the hypotenuse of the triangle spanning the two apexes = \sqrt{3^2 + 3^2}. The angle \theta will be maximized when cos\theta is minimized.

    So simply differentiate the function \frac{a^2+b^2-c^2}{2ab} with respect to P to find the minimum.
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