SOLUTION: A farmer decides to enclose a rectangular field using the side of a barn as one side of the rectangle. The figure below shows the fenced area he wants to make. If he has 2800 fe

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Question 615351: A farmer decides to enclose a rectangular field using the side of a barn as one side of the rectangle. The figure below shows the fenced area he wants to make.
If he has 2800 feet of fencing, what values of x and y will maximize the enclosed area?
x = ? feet
y = ? feet
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = the lengths of the sides of the field that are perpendicular to the barn.
Let y = the length of the side of the field that is parallel to the barn.

With x and y we can now express the perimeter of the field:
x + x + y = 2800
which simplifies to
2x + y = 2800

The area of the field will be:
A = x*y

TO solve this we start by solving the perimeter equation for y. Subtracting 2x from each side we get:
y = 2800 - 2x

We can substitute this into the area equation for the y:
A = x*(2800 - 2x)
which simplifies to:
A+=+2800x+-+2x%5E2
or
A+=+-2x%5E2+%2B+2800x

Assuming you are not in a Calculus class, the next step is to recognize that the Area equation is the equation of a parabola. And, since the coefficient of x%5E2 is negative, this parabola opens downward. If we picture such a parabola, we should be able to realize that it will have a maximum (or highest) point. This will be the vertex of the parabola. So we want to find the x coordinate for the vertex of the parabola.

You may have learned that the x coordinate of the vertex of a parabola is equal to -b%2F%282a%29. Since the b = 2800 and the a = -2, the x coordinate of our parabola will be:
x+=+-%282800%29%2F%282%28-2%29%29
which simplifies as follows:
x+=+-%282800%29%2F%28-4%29
x = 700

So the value for x that gives us the greatest area is 700. We can use
y = 2800 - 2x
to find the y value:
y = 2800 - 2(700)
y = 2800 - 1400
y = 1400

So the maximum area can be achieved if the sides perpendicular to the barn are 700 feet and the side parallel to the barn is 1400 feet.

P.S. This solution means the barn must be 1400 feet long itself in order for it to be the 4th side of the field. That's an awfully long barn!!