Question

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=7-x^2 .

Answer 1

If the base is on the x-axis, the width of the rectangle is = xif the height is based on the parabola, the length = 7-x^2
the area of a rectangle = length * width
thus, area = x*(7-x^2)= 7x-x^3
in order to maximize the area, you would need to take the derivative of the area and set it equal to 0
Area = 7x-x^3Area' = 7 - 3x^2
7-3x^2 = 0thus, x = 1.5275
this x represents the x needed to create the largest possible area with the given parameters.
Thus:Width (x-axis) = 1.5275Length (y-axis) = 7 - (1.5275)^2 = 4.6667