# Maximization

#### cdlegendary

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

The angles around theta add up to 180 (they are on a straight line)

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

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

Can you differenciate that? looks like it might get messy...

• cdlegendary

#### GeoC

use $$\displaystyle a^2 + b^2 - c^2 = 2abcos\theta$$ with $$\displaystyle a$$ the length of the hypotenuse defined by $$\displaystyle \sqrt{(3-P)^2 + 2^2}$$; $$\displaystyle b$$ the length of the hypotenuse defined by $$\displaystyle \sqrt{P^2 + 5^2}$$; and $$\displaystyle c$$ the hypotenuse of the triangle spanning the two apexes = $$\displaystyle \sqrt{3^2 + 3^2}$$. The angle $$\displaystyle \theta$$ will be maximized when $$\displaystyle cos\theta$$ is minimized.

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