Arectangular plot of farmland will be bounded on one side by a river and on the other three sides by aâ single-strand electric fence. with 500500 m of wire at yourâ disposal, what is the largest area you canâ enclose, and what are itsâ dimensions?

If one side is bounded by the river, the 500m of wire must make the remain three sides of the rectangular plot of land;
Since it is a rectangle, you will have two pairs of equal and parallel sides;
So, we will have to make 2 parallel and equal lengths of wire and one other length parallel to the river;
If we let:
x = the side that is perpendicular to the river
y = the side parallel to the river
We can say:
2x + y = 500 [1]
We can also say for the area of the land plot:
A = xy [2]
Now, we rearrange [1] and insert it into the [2]:
y = 500 - 2x
A = x(500 - 2x)
= -2x² + 500x
Now, we differentiate our equation for the area of the land, set it equal to 0 and solve for x (notice that the equation we get for the area is a negative x² graph and so we know it will be rainbow-shaped, which means it has a highest point, this will represent the maximum area):
dA/dx = -4x + 500 = 0
4x = 500
x = 125
We can now plug this x-value into our equation [1] to find y:
y = 500 - 2x
y = 500 - 2(125)
= 500 - 250 = 250
This means the largest area you can obtain would have two lengths of 125m that are perpendicular to the river and a length of 250m opposite and parallel to the river.
The area (which is the largest possible with these dimensions) can be found by plugging in the x and y values we have into equation [2] or by plugging just the x-value into the other equation for area we formulated, i.e.
A = -2x² + 500x