Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola.

Question

y = 7 − x²

^width:

^height:

Answer 1

y=7-x^2 is a downward opening parabola

with vertex= maximum = (0,7)

find the maximum area of a rectangle with base = y=0 (the x axis) and top two vertices = points on the parabola

Area of half the rectangle = xy where (x,y) is one vertex of the rectangle

xy = x(7-x^2)= 7x -x^3

take the derivative of A=7x-x^3

A' =7 -3x^2 = 0

x^2= 7/3

x = sqr(7/3)

max rectangle Area = 2A =14x-2x^2x = 14sqr(7/3) -2(7/3)sqr(7/3)

= (28/3)sqr(7/3)= about 14.26 square units

Answer 2

The area of the largest rectangle shall be

A = 2(x(7 - x²))

Differentiate and determine length and width.