SOLUTION: Faye has 20 feet of fencing to make a rectangle pen for her dog. what is the largest area that she can fence in?

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Question 516683: Faye has 20 feet of fencing to make a rectangle pen for her dog. what is the largest area that she can fence in?
Answer by solver91311(24713) About Me  (Show Source):
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The rectangle with the greatest area for a given perimeter is a square with sides that measure the perimeter divided by 4.

The perimeter of a rectangle is given by:



The area of a rectangle is given by:



Substituting from the perimeter equation:



This function graphs to a parabola opening downward meaning that the vertex is a maximum.

Algebra solution: The vertex is the maximum, so for a parabola with equation , the vertex is at . For this parabola:

Calculus solution: The maximum value of the function, hence the maximum area, is where the value of the first derivative is equal to zero:



Set equal to zero:



Hence, the maximum area rectangle for a given perimeter is a square with sides of length one-fourth of the perimeter.



John

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