A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 11 − x 2 . What are the dimensions of such a rectangle with the greatest possible area?

Question

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola . What are the dimensions of such a rectangle with the greatest possible area?

Width:

Height:

Answer 1

The rectangle will have width 2x and height 11 - x², so A = wh = 22x - 2x³.

Thus, we maximize A by setting dA/dx = 0

d(22x - 2x³)/dx = 22 - 6x²

22 - 6x² = 0

x² = 11/3

x = ±√(11/3) or ±√33 / 3

h = 11 - x² = 11 - 11/3 = 22/3

the base of the rectangle is 2√33 / 3 The height is 22/3

The area is 44√33 / 9