Answer
Maximum area A_max = 11250 ft^2
Step-by-Step Explanation
Declaring Variables:-
The length of the rectangle = y
The width of the rectangle = x
Solution:-
- The perimeter of a rectangle can be expressed using the above two variables as follows:
               Perimeter (P) = 2*Length + 2*Width
                           = 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. The length of the fence required L is:
              Length (L) = P - y
                        = 2*x + y
- We are given 300 feet of fencing. So we equate the length equal to 300 and develop a linear relationship between width and length of the barn.
               300 = 2x + y
               y = 300 - 2x
- The area (A) of the rectangle is given by the following expression:
               A = Length*width
               A = x*y
- Substitute the relationship developed between x and y in the Area (A) expression above. Then we have:
               A = x*(300 - 2x)
               A = 300x - 2x^2
- We will take first derivative of the expression of area (A) developed with respect to x and find the critical point of the area function by setting the first derivative A'(x) = 0.
               A(x) = 300x - 2x^2
               A'(x) = 300 - 4x
               0 = 300-4x
               x = 300 / 4 = 75 ft
- The critical point of the given function lies for the width (x) of 75 ft. We will plug in the critical value x = 75 ft back into the original function of Area and find the maximum area.
               A(x) = 300x - 2x^2
               A(75) = 300 (75) - 2(75)^2
               A_max = A(75) = 11250 ft^2
- The maximum area that can be enclosed by the fencing is 11250 ft²
