Calculus Optimization Problem - Need Serious Help

[Kirobaito]

Okay, we're going over optimization in Calculus right now. I understand it perfectly when my teacher does it on the board; I know exactly what he's doing and why. But every time he gives us a problem to work on our own I get lost and end up going in circles and the like. We got one, and I've tried it a hundred times, and can never make it easier. Help would be appreciated. My work so far is shown.

AP CALCULUS
FIND THE DIMENSIONS OF THE LARGEST ISOSCELES TRIANGLE THAT CAN BE INSCRIBED IN A CIRCLE OF RADIUS 4.

OK, so pretty much this is how I've drawn it up.

<b>Okay, Photobucket is being a meanie, so it's now attached.</b>


Now, here are my formulas that I've made. I'm attempting to make the Area formula contain only one variable, in this case x.

<b>A=xy</b>
The formula for the area in this problem, since the base is actually 2x. Makes it easier.

<b>y=4+h</b>
Simple logic.

h^2+x^2=16
<b>h=(16-x^2)^(1/2)</b>
OK, the formula for h in terms of x, so I can replace it in the area formula. I used the pythagorean theorem.

<b>A=x[4+(16-x^2)^(1/2)]</b>
OK, this is the area formula, and it only contains X. Now, before I find the derivative in terms of X so I can find the Maximum of the A(x) curve, I'm going to distribute the x.

<b>A=4x+(x)(16-x^2)^(1/2)</b>
I'm correct so far, yes?

Now, here's where it all gets messy. I'm going to find the derivative in terms of X, and then set dA/dx equal to zero to find the maximum.

<b>A=4x+(x)(16-x^2)^(1/2)</b>
dA/dx=4+....
OK, now here's me using the product and chain rules.
[(1)(16-x^2)^(1/2)+x(1/2)(16-x^2)^(-1/2)(0-2x)]

That simplifies to:

<b>dA/dx=4+[16-x^2)^(1/2)+x(1/2)(16-x^2)^(-1/2)(-2x)]</b>

Now, the 1/2 and the 2 in the 2x at the end can cancel. So make it:

<b>dA/dx=4+[16-x^2)^(1/2)+x(16-x^2)^(-1/2)(-x)]</b>

I'm gonna combine the -x and x now to make -x^2.

<b>dA/dx=4+(16-x^2)^(1/2)+(-x^2)(16-x^2)^(-1/2)</b>

Here's where it gets so messy it begins difficult to type.

<b>0=4+(16-x^2)^(1/2)+[(-x^2)/[(16-x^2)^(1/2)]]</b>

Does that make sense? Now, I'm gonna multiply every value by (16-x^2) so I can get it out of the denominator on the final term. It would help if you had been writing this, if anybody chose to help me.

<b>0=4(16-x^2)+(16-x^2)[(16-x^2)^(1/2)]-x^2

Is this correct AT ALL?

The work gets way to messy and it's here that I start to doubt myself. Have I done any of this right?