A rectangle with its base on the x-axis is to be inscribed under the graph of y=2-x^2. Find the height of the rectangle if the area is the largest possible area. (Round to the nearest hundreth)

Answer 1

Let the point of contact of the rectangle in the first quadrant be (x,y)

then the base of the rectangle is 2x and the height is y
Area = 2xy
= 2x(2-x^2)
= 4x - 2x^3
d(Area)/dx = 4 - 6x^2
= 0 for a max of Area
6x^2 = 4
x^2 = 4/6
x = ±2/6
then height = y = 2 - 4/6 = 4/3
= 1.33 to the nearest hundreth