Find the dimensions that give the largest area for the rectangle shown in the figure.

Question

Its base is on the x-axis and its other two vertices are above the x-axis, lying on the parabola y = k − x2, k = 8.

Answer 1

y= 8-x^2

(x,y) = (x,8-x^2)

area of half the rectangle on the right side, in Quadrant I

= xy = x(8-x^2) = 8x -x^3 = Area = A

A' = 8-3x^2 = 0

3x^2 = 8

x^2 = 8/3

x = sqr(8/3) = about about 1.6

y = 8-x^2 = 8-8/3 = 24/3-8/2 = 16/3 = 5 1/3

base of the full rectangle = 2x = 2sqr(8/3)

height of the full rectangle = 16/3

largest area = 2xy = 2sqr(8/3)(16/3) = (32/3)sqr(8/3)

= about 17.4

vertices of the rectangle are (sqr(8/3), 16/3), (-sqr(8/3),16/3), (sqr(8/3), 0), (-sqr(8/3), 0)