A farmer decide to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the framer can enclose with 120 ft of fence? What should the dimension of the garden be to given this area?
Picture the bran and the garden as in the following figure.
A formula for the use of the fencing is 120 = 2W + L.
A formula for the area of the garden A = LW.
Now, you must express A as a function of L or W.
Find A as a function of W. To do this, you must solve 120 = 2W + L for L. This gives
L = 120 – 2W.
Substituting for L in the equation for area, given A = (120-2W)W.
To find the maximum function value, you must complete the square.
A = 120W – 2W2
A = -2(W2 – 60W)
A = -2(W2 – 60W + 900) + 1800
A = -2(W – 30)2 + 1800
This tells you that the maximum function value, or the maximum area, is 1800 sq ft.
The maximum area is obtained when the width is 30 ft and the length is 60 ft.