# Using differentiation to find maximum length problems

1. Feb 14, 2013

### lmstaples

1. The problem statement, all variables and given/known data

The line segment AB lies on a diameter of a circle of radius 1, and the angle BAC is a right angle.

Find the greatest possible value of the sum of the lengths of AB and AC.

2. Relevant equations

3. The attempt at a solution

I have no idea what parameters to use or how to set up the problem.

Any help would be much appreciated.

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2. Feb 14, 2013

### Dick

Why not try picking the parameter to be the x coordinate of A?

3. Feb 14, 2013

### lmstaples

check out the attachment, does that seem right?

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4. Feb 14, 2013

### Dick

Well, you ignored my advice on which parameter to use, so apparently you weren't all that confused to begin with. Yes, it looks ok.

5. Feb 14, 2013

### lmstaples

Haha sorry, I had so many ideas of ways to do it and that's the only one I managed to fully work through :)

6. Feb 15, 2013

### HallsofIvy

Staff Emeritus
Here (I think) is what Dick was suggesting: set up a coordinate system so the origin is at the center of the circle, A is at (-1, 0), and the y-axis is parallel to BC. Then AB has length x+ 1 and BC has length $\sqrt{1- x^2}$.