SOLUTION: A rectangular banner is to have a red border and a rectangular white center. The width of the border is to be 8 inches along the top and bottom and 6 inches along the sides. The to

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Question 999498: A rectangular banner is to have a red border and a rectangular white center. The width of the border is to be 8 inches along the top and bottom and 6 inches along the sides. The total area is to be 27 square feet. Find the dimensions of the banner that will maximize the area of the white center.
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!

x = width of the banner
and y = length
therefore xy = 27
Area of centre,

A = (x - 1)(y - 16/12)
= (x - 1)(27/x - 4/3)
= 27 - (4/3)x - 27/x + 4/3
= 85/3 - (4/3)x - 27/x
For f(x)=85/3 - (4/3)x - 27/x , to be maximum,
f '(x) = 0 and f "(x) < 0
f '(x) = 0
= -4/3 + 27/x^2 = 0
= x^2 = 81/4
= x = 9/2

f "(x) = - 54/x^2 < 0
x = 9/4 => y = 27 * 4/9 = 12
length and width of banner = 12 ft. x 2 1/4 ft.
12*2 1/4 = 27