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) (Show Source): You can put this solution on YOUR website!
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:
 = w\left(\frac{P}{2} - w\right)\ \ \Rightarrow\ \ A(w) = \left(\frac{wP}{2} - w^2\right))
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 \ =\ ax^2\ +\ bx\ +\ c)
, the vertex is at \right))
. For this parabola: },f\left(\frac{-\frac{P}{2}}{2(-1)}\right)\right)\ =\ \left(\frac{P}{4},f\left(\frac{P}{4}\right)\right))
Calculus solution: The maximum value of the function, hence the maximum area, is where the value of the first derivative is equal to zero:
\ =\ -2w + \frac{P}{2})
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
My calculator said it, I believe it, that settles it
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