Question

you have 160 yards of fencing to enclose a rectangular region. find the maximum area of the rectangular region

Answer 1

I know that for a fixed perimeter, the shape that encloses the greatest area
is a circle, and the rectangle that encloses the greatest area is a square. 
Sadly, I don't know how to prove it to you without Calculus.

If you'll take my assertion that the greatest rectangle is a square, and accept
it on faith, then you should use your 160-yd of fence to enclose a square with
40-yd sides.  The area inside it is (40 x 40) = 1600 square yards.

Here are some other choices. 
Each one has the same perimeter ... 160 yards.
This table kind of suggests to you that a square is the rectangle
       with the greatest area.  (But it doesn't prove it.)

Length         Width         Area
  35-yd           45-yd         1,575 square yards              
  30                50              1,500
  25                55              1,375
  20                60              1,200
  15                65                 975
  10                70                 700
    5                75                 375
    3                77                 231
    2                78                 156
    1                79                   79
  2-ft           79-yd 1-ft           52.89
  1-ft           79-yd 2-ft           26.56
  1-inch      79-yd 35-in          2.19 square yards

With the same 160-yd of fence, you could have squeezed in some more
area by setting the fence down in a circle with circumference = 160-yd.
The area inside the circle would be

       Area = (pi) (radius)² = (circumference)² / (4 pi) = 2,037.2 square yards

The area of a circle is always (4 / pi) times the area of the square with the
same perimeter.  That's about 27.3% more area than the square.

Answer 2

To optimize the area, the shape should be a square.  Clearly, with 160 yards of fencing, the square will have side-length 160/4=40 yards.  Therefore, the area will be 40*40=1600 yards.