Let's say the length of sides perpendicular to the river = x
and, the length of side parallel to the river = y
Since the side along the river is not fenced, only 3 sides remain to be fenced.
2x + y = 8500 eq1
Area of the rectangle = x*y
substitute value of y from eq1
area = x(8500-2x)
=-2x2 + 8500x eq2
Since the coefficient of x2 is negative, this is a parabola opening down,
in which case it will have it's maximum value at the vertex.
Now, the vertex of parabola with standard form f(x) = ax2 + bx + c is at x=-b/2a
so, the vertex of parabola represented by eq2 will be at x = -8500/2(-2) = 2125
This means that area will be maximum when x = 2125
We plug this value in eq2 to get the maximum area.
Maximum area = -2(2125)2 + 8500(2125) = 9,031,250 meters2