The base of a rectangle is on the x-axis and its two upper vertices are on the parabola y=25-x^2. of all such rectangles, what are the dimensions of the one with greatest area?

Answer 1

Clearly the maximum area is when the rectangle is centered on the parabola's axis of symmetry. In this case, x=0. So, let the base of the rectangle be of length 2x. Its area is thus

A = 2xy = 2x(25-x^2) = 50x-2x^3
dA/dx = 50-6x^2
dA/dx=0 when x=5/√3

So, the maximum area is (10/√3)(25-(5/√3)^2) = 500/(3√3)