Maximization

**cdlegendary** · May 31st 2010, 02:12 PM

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.

**SpringFan25** · May 31st 2010, 02:38 PM

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...

**GeoC** · May 31st 2010, 02:47 PM

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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