Farmer Ed has

9 comma 0009,000
meters of​ fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does not fence the side along the​ river, what is the largest area that can be​ enclosed?

What is this?

9 comma 0009,000

if the side parallel to the river is x, and the other two sides are y, you have

x+2y = 9000
the area
a = xy = (9000-2y)(y) = 9000y-2y^2
The vertex of this parabola is where you have maximum area. That is at y = 9000/4

You have y, so you can get x, and can find the area.

10125000

To find the largest area that can be enclosed by Farmer Ed's rectangular plot, we need to determine the dimensions of the rectangle that will maximize the area.

Let's call the length of the rectangle l and the width w. Since the rectangle borders on a river, only three sides will be fenced (two widths and one length).

The perimeter of the rectangle can be calculated by summing the lengths of the three sides:
Perimeter = 2w + l

Given that Farmer Ed has 9,000 meters of fencing, we can write the equation:
2w + l = 9,000

To find the largest area, we need to maximize the value of lw (length times width).

Now we will use the perimeter equation to express one variable in terms of the other and substitute it into the area equation, transforming it into a single-variable equation.

From the perimeter equation, we can solve for l:
l = 9,000 - 2w

Substituting this value of l into the area equation:
Area = lw = (9,000 - 2w)w

Now, we have a quadratic equation for the area in terms of w. To find the maximum area, we need to take the derivative of the area equation with respect to w, set it equal to zero, and solve for w.

Let's differentiate the area equation with respect to w:
d(Area)/dw = 9,000 - 4w = 0

Simplifying the equation:
9,000 = 4w

Solving for w:
w = 9,000 / 4 = 2,250

Now that we have the value of w, we can substitute it back into the perimeter equation to find the corresponding value of l:
l = 9,000 - 2w = 9,000 - 2(2,250) = 9,000 - 4,500 = 4,500

Therefore, the largest area that can be enclosed by Farmer Ed's rectangular plot is given by a width of 2,250 meters and a length of 4,500 meters.