Maximization

Jan 2010
79
1
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.
 
May 2010
1,034
272
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...
 
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May 2010
43
1
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.