how big can the rectangle be?

Question

 A parabola has a vertex at (0,100) and x-intercepts at (-50, 0) and (50, 0). What is the maximum area of a rectangle drawn between the parabola and the x-axis?

Answer 1

parabola equation is: y= -1/25(x^2) + 100

Area = A' = xy =- x^3/25 + 100x

the area = integral

take the derivative & set = 0

A' = - 3x^2/25 + 100 = 0

x^2 = 100(25)/3

x = 50sqr3/3= 28.9

y= -1/25(100)(25/3) + 100 = 66.7

xy = 28.9(66.7) = 1928 for half the area

total area = 3855 for the rectangular maximum area