To find the largest area that can be enclosed, we need to determine the dimensions of the rectangular plot.
Let's assume the length of the rectangular plot is L and the width is W.
Given that Farmer Ed has 9,000 meters of fencing, we can calculate the perimeter of the rectangular plot:
Perimeter = 2L + W
Since we are not fencing the side along the river, the perimeter will be:
Perimeter = L + 2W
We also know that the perimeter is equal to 9,000 meters:
L + 2W = 9,000
Now, we need to express one of the variables in terms of the other so that we can maximize the area.
Rearranging the equation, we get:
L = 9,000 - 2W
The area of a rectangle is given by:
Area = Length x Width
Substituting the expression for L into the area formula, we get:
Area = (9,000 - 2W) x W
Now, we need to find the value of W that maximizes the area. We can do this by finding the maximum of the quadratic equation.
To find the maximum of the quadratic equation, we can calculate the vertex using the formula:
W = -b / (2a)
In this case, a = -2 and b = 9,000.
W = -9,000 / (2*(-2))
W = -9,000 / (-4)
W = 2,250
We can now find the value of L by substituting W back into the equation:
L = 9,000 - 2(2,250)
L = 9,000 - 4,500
L = 4,500
Therefore, the width is 2,250 meters and the length is 4,500 meters.
Finally, we can calculate the maximum area by multiplying the width and length:
Area = 2,250 x 4,500
Area = 10,125,000 square meters
So, the largest area that can be enclosed is 10,125,000 square meters.