The fencing problem is a problem used to explain how the quadratic function is used to find the dimensions of a rectangular yard that will produce the greatest area.
Example: A fencing of 2000 feet is used to create a rectangular yard with one side of the yard being the wall. Create a rectangular region with the 2000 feet fencing representing the other three sides of the rectangular region. The picture below illustrates the problem:
Let A(x) = Area of the rectangular region in terms of x.
A(x) = (L)(W)
A(x) = (2000 - 2x)x = 2000x - 2x^2
A(x) = - 2x^2 + 2000x
a = - 2 < 0 means the graph of the function is concave down
Line of symmetry passes through the vertex with the equation of the line of symmetry:
x = -b/(2a)
x = - 2000/(2(-2)) = 500 feet.
Because the vertex for this function is at the maximum value of the function, x = 500 feet represents the width that would give the greatest area. Therefore, L = 2000 - 2(500) = 1000 feet.
The dimensions of the rectangular region that will give the greatest area is:
L = 1000 feet
W = 500 feet
Maximum Area = 1000(500) = 500,000 square feet