Let
y------> the length of the rectangle
x-----> the width of the rectangle
we know that
perimeter of the rectangle=2*[x+y]
Since the barn is used as one of the sides (let's say y) we can subtract y
we don't need fencing for this side
That makes the perimeter 2x + y
Since we have 300 feet of fencing
we set these equal:
2x + y = 300-------> y=(300-2x)------> equation 1
area of the rectangle=x*y
substitute the equation 1 in the area formula
Area=[(300-2x)]*x-----> Area=(-2x²)+300x
This is a quadratic equation
Since the leading coefficient is negative (-2) we know it opens downward
We are looking for the x-coordinate of the highest point called the vertex
using a graph tool
see the attached figure
the vertex is the point (75,11250)
that means for x=75 (width of the rectangle)
the area is 11250 ft²
the answer is
the maximum area that can be enclosed by the fencing is 11250 ft²
