SOLUTION: A homeowner wants to fence a rectangular garden using 120ft of fencing. The side of the garage will be used as one side of the rectangle. Find the dimenstions for which the area

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Question 162645: A homeowner wants to fence a rectangular garden using 120ft of fencing. The side of the garage will be used as one side of the rectangle. Find the dimenstions for which the area is a maximum.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A homeowner wants to fence a rectangular garden using 120ft of fencing. The side of the garage will be used as one side of the rectangle. Find the dimensions for which the area is a maximum.
:
The perimeter equation with one side being the garage:
L + 2W = 120
or
L = (120 - 2W)
;
Area = W * L
Substitute (120-2W) for L and you have:
A = W(120-2W)
A = 120W - 2W^2
This can be arranged as a quadratic equation
-2W^2 + 120W = 0
:
To find the maximum, find the axis symmetry using x = %28-b%29%2F%282%2Aa%29
In this equation:
W = %28-120%29%2F%282%2A-2%29
W = %28-120%29%2F%28-4%29
W = +30 ft is the width for max area
Find L
L = 120 - 2(30)
L = 60 ft is the length
;
Find the max area, substitute 30 for W in the equation
A = -2(30^2) + 120(30)
A = -1800 = 3600
A = +1800 sq/ft is the maximum area
:
:
Check solution by finding the area with the dimensions:
60 * 30 = 1800 sq/ft