Width and height

Question

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?

Width =   

Height =   

Answer 1

Draw a diagram. Coordinates of upper right corner of the rectangle are (x, 11-x²).

width of rectangle = 2x and height = 11 - x²

Area of rectangle = A = 2x(11 - x²) = 22x - 2x³

A' = 22 - 6x² = 0 when x = √(11/3)

Dimensions: width = 2x = 2√33 / 3 and height = 11 - 11/3 = 22/3