Question

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?
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)

Related Questions

Applications of derivatives A rectangle has its base on the x axis and its upper two vertices on... A rectangle has its base on the x-axis and its upper two vertices on the parabola y=12−x2 . Wha... What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the cu...