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) = x* (2100 - x ) /2 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
y = ( 2100 - x ) /2 then
y = ( 2100 - 1050 ) / 2
y = 525 yd
And
A(max) = 1050* 525
A(max) = 551250 yd²
