To find the dimensions of the region with the largest area, we need to consider the perimeter of the fence and how it relates to the area of the region.
Let's assume the length of the region is L and the width is W.
Given that three sides will require fencing and one side is the existing wall, the perimeter of the fence will be:
Perimeter = 2L + W
We know that the farmer has 180 feet of fencing, so we can write the equation:
2L + W = 180
To find the dimensions of the region with the largest area, we need to find the maximum area. The area of the region is given by:
Area = L * W
To maximize the area, we can solve for one variable in terms of the other and substitute it into the area equation.
From the perimeter equation, we can solve for W:
W = 180 - 2L
Substituting this value into the area equation:
Area = L * (180 - 2L)
To find the maximum area, we need to find the value of L that maximizes the area. One way to do this is by graphing the area equation and finding the highest point on the graph.
However, since we are looking for a clear and concise answer, we can find the maximum area by completing the square.
Rearranging the area equation:
Area = -2L^2 + 180L
To complete the square, we can rewrite this equation as:
Area = -2(L^2 - 90L)
To complete the square, we need to add and subtract half of the coefficient of L squared, which is (90/2)^2 = 4050, inside the parentheses:
Area = -2(L^2 - 90L + 4050 - 4050)
Simplifying:
Area = -2((L - 45)^2 - 4050)
Expanding:
Area = -2(L - 45)^2 + 8100
From this equation, we can see that the maximum area occurs when (L - 45)^2 = 0, which means L = 45.
Substituting L = 45 into the perimeter equation:
2(45) + W = 180
90 + W = 180
W = 90
Therefore, the dimensions of the region with the largest area are 45 feet by 90 feet.