When I was in school, Bill Gates was just born, so we did not have Excel yet. In other words, if you wanted a wrong answer, it took a lot longer to get it; today, just open Excel, and you can be wrong really fast.
So, I will solve this the "old-fashioned way". We let "x" be half the base of the rectangle -- the base goes from -x to +x, totaling 2x in length. The height is "y", and y is on the parabola, y = 9 - x2. Therefore the area of the rectangle is
A = 2x * y
= 2x * (9 - x2)
= 18x - 2x3
And now the old-fashioned math: to find the maximum, take the derivative, set it =0, solve, and make sure the solution is a maximum not a minimum.
Thus, the derivative dA/dx = 18 - (2 * 3 * x2) = 18 - 6x2
Setting dA/dx=0 gives us
18 - 6x2 = 0
so, 18 = 6x2
so, 3 = x2
so, x = ±√3
We have defined "x" as the length of half the base, so our solution is the positive root, x=√3. To check if this solution is a max/min, we take the second derivative -- d2A/dx2 = -2*6*x1 = -12x. Therefore, at x=√3, the second derivative is d2A/dx2 = -12√3, which is negative. Therefore our solution is a maximum, and we have solved the problem. Note that for x=√3, y = 9-x2 = 6
To summarize, the maximum area is at (x,y) = (√3, 6), corresponding to a rectangle with base of 2x = 2√3, and height of 6. The area is (2√3 * 6) = 12√3 ≈ 20.78
Wow - we solved it without Excel, and got the correct answer.
I hope this has helped you out.