SOLUTION: A rectangular field is to be enclosed by 440 feet of fence. What is the length of the field (in feet) if the area is a maximum?
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Question 1181394: A rectangular field is to be enclosed by 440 feet of fence. What is the length of the field (in feet) if the area is a maximum? Found 2 solutions by greenestamps, ikleyn:Answer by greenestamps(13203) (Show Source):
A rectangle with fixed perimeter has maximum area when the rectangle is a square.
Given 440 feet of fencing, the maximum area is when the length and width are both 440/4=110 feet.
Here is a quick way to show this algebraically....
The perimeter is 440 feet, so length plus width is 220 feet.
To make the sum of length and width equal to 220, let the length be 110+x and the width be 110-x. Then the area is length times width: (110+x)(110-x)=12100-x^2.
x^2 is always 0 or positive; so the maximum area is when x is 0, making the length and width both 110 feet.
