The figure below shows the curves y=square root of x, x=9, y=0 and a rectangle with the sides parallel to the axes and its left end at x=a. Find the dimensions of the rectangle having the maximum possible area.

Answer 1

If the lower left corner is at x, the height is sqrt(x) and the width is (9-x)

a = 9x^1/2 - x^3/2

a' = 9/(2x^1/2) - 3/2 x^1/2
= 3/(2x^1/2) * (3 - x)

a' = 0 at x=3

a(3) = sqrt(3) * (9-3) = 6sqrt(3)