1. ## Optimization help?

Hello all. I am having trouble with a couple of optimization problems and was wondering if anyone could help.

What point on the graph of y = √x is closest to the point (3,0)? *note: That is y = the square root of x*

A boat leaves a dock at 2:00 p.m. and travels due south at a speed of 20 mph. Another boat which had been heading due east at 15 mph, arrives at the dock at 3:00 p.m. At what time were the two boats closest together?

Assume you draw a line segment of length 20 units that starts on the positive x-axis and ends on the positive y-axis. That line, together with the x-axis and y-axis, forms a triabgle. What is the maximum possible area of such a triangle?

If you all can help with any of these, I would be super grateful!

2. Originally Posted by stockha0
What point on the graph of y = √x is closest to the point (3,0)? *note: That is y = the square root of x*
You must think your way through them.

Points on $\displaystyle y = \sqrt{x}$ look like this: $\displaystyle (x,\sqrt{x})$

Knowing that, such points are what distance from (3,0)? Perhaps $\displaystyle \sqrt{(x-3)^{2}+[\sqrt{x}-0]^{2}}\;=\;\sqrt{(x-3)^{2}+x}$

The real art of this problem is realizing that it does not qute matter that you actually minimize the distance directly. Since the square root is so well-behaved, you can just as effectively minimize the square of the distance. It is the beauty of a unique answer. It doesn't care how you find it.