A farmer has 640 feet of fencing. He plans to enclose a rectangular region bordering a
River (with no fence along the river side). What dimensions should he use to have an
Enclosure of the largest possible area?

Since there are only three sides, the perimeter is P=x+2y
But this is equal to 640.

So, we have x+2y=640

The area is A=xy

Solve [1] for y (or x) and sub into A. It is now in terms of one variable.

You can differentiate, or use

x=\frac{-b}{2a}, \;\ y=c-\frac{b^{2}}{4a} to find the optimum dimensions.