Question 615351
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.<br>
With x and y we can now express the perimeter of the field:
x + x + y = 2800
which simplifies to
2x + y = 2800<br>
The area of the field will be:
A = x*y<br>
TO solve this we start by solving the perimeter equation for y. Subtracting 2x from each side we get:
y = 2800 - 2x<br>
We can substitute this into the area equation for the y:
A = x*(2800 - 2x)
which simplifies to:
{{{A = 2800x - 2x^2}}}
or
{{{A = -2x^2 + 2800x}}}<br>
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^2}}} 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.<br>
You may have learned that the x coordinate of the vertex of a parabola is equal to {{{-b/(2a)}}}. Since the b = 2800  and the a = -2, the x coordinate of our parabola will be:
{{{x = -(2800)/(2(-2))}}}
which simplifies as follows:
{{{x = -(2800)/(-4)}}}
x = 700<br>
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<br>
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.<br>
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!!