A rectangle is inscribed under the curve y = 24 – x2 with a portion of the x-axis as its base. What is the area, in square units, of the largest such rectangle?

Answer: 90.51 square units

Explanation:

1) The rectangle has one length on the x-axis.

2) Call x, the x-coordinate of the right lower corner of the rectangle.

And the point is (x,0)

3) The x-coordinate of the lef lower corner will be - x.

And the point is (-x,0)

4) The right upper point of the rectangle will be (x,y) where y = 24 - x²

5) The left upper point of the rectangle will be (-x,y) where y = 24 - x² .

6) The area of such rectantle is the length of the base times the length of the height.

The length of the base is 2x

The lenght of the height is y

So, the area is A = 2x(y) = 2x(24 - x²) = 48x - 2x³

7) Now to find the maximum area you have to derivate the function and make it equal to zero:

dA
---- = A' = 48 - 6x² = 0 =>
dx

4x² = 48 => x² = 48/6 = 8 =>

x = (+/-)√8

Then, y = 24 - x² = 24 - (√8)² = 24 - 8 = 16

8) Therefore the area of the rectangle is

A = 2x(y) = 2(√8)(16) = 90.51.

9) You can see if x is a greater or smaller.

For example x = 2 and x = 3

x = 2 => y = 24 - (2)² = 20 => A = 2(2)(20) = 80 which is less than 90.51

x = 3 => y = 24 - (3)² = 15 => A = 2(3)(15) = 90 which is less than 90.51

So, that tells you that your result should be right.