a calculus optimization problem

Question

Given points A(2,1) and B(5,4), find the point on the x-axis P(x,0) in the interval [2,5] that maximizes the angle APB.

How can I devise an optimize equation and a constraint equation out of this?

Use the $\arctan$ function. – Tony Piccolo Nov 21 '14 at 15:03 — Tony Piccolo, Nov 21 '14 at 15:03

score 0 · Answer 1 · answered Nov 21 '14 at 06:40

0

Consider points A$(2,1)$ and P$(x,0)$ ,

Then, Vector $\vec {PA}$ = $(x-2)\hat{i}+(-1)\hat{j}$

And similarly P$(x,0)$ and B$(5,4)$

Vector $\vec {PB}=(x-5)\hat{i}+(-4)\hat{j}$
And $\cos{\theta}=\large\frac{\vec {PA}.\vec {PB}}{\|PA\|.\|PB\|}$

I assume you can continue from here! Good luck!

answered Nov 21 '14 at 06:40

Someone

Do we really have to use vectors for this? The student I'm tutoring (who is asking me this problem) is only in Calculus 1. Can we avoid vectors please? – user194325 Nov 21 '14 at 06:50
Uhm, then maybe you can use the identity. $\cos C=\large{\frac{c^2-a^2-b^2}{2ab}}$ – Someone Nov 21 '14 at 07:31

1 Answers1