Answer:
x  = 1050 yd
y  = 525  yd
A(max)  =  551250 yd²
Step-by-step explanation:
For enclosing a rectangular area (only three sides, since one side will be the river) we have 2100 yards, then the length of fencing material is:
L  =  2100 = x + 2y   ⇒  y  =  ( 2100  -  x  ) / 2
Where x and y are the sides of the rectangle ( x is the parallel side to the river)
The area of the rectangle is:
A = x*y
And as    y = (2100 - x ) / 2
We can express  A as a function of x, getting:
A(x) Â = Â x* (2100 Â - x ) /2 Â Â Â or
A(x)  =( 2100*x  - x² )/ 2    ⇒ A(x)  =  1050*x  - (1/2)*x²    (1)
Taking derivatives on both sides of the equation we have
A´(x)  =  1050 - x
A´(x)  =  0  means    1050 - x  = 0
x  =  1050 yd
And as A´´(x)  =  - 1    A´´(x) < 0
We have a maximum for the function at the point x = 1050
Now Â
y  = ( 2100  -  x ) /2    then
y  = ( 2100 -  1050 ) / 2
y  = 525 yd
And
A(max) Â = Â 1050* 525
A(max) =  551250 yd²
