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
