A rancher has 600 feet of fencing to put around a rectangular field and then subdivide the field into 22 identical smaller rectangular plots by placing a fence parallel to one of the field's shorter sides. find the dimensions that maximize the enclosed area. write your answers as fractions reduced to lowest terms.

The fence shape would be 2 identical smaller rectangular plots by placing a fence parallel to one of the field's shorter sides.

______
|             |
|______|   long=L
|             |
|______|
short= S

From the pic, you can see that you need to build 3 short fences and 2 long fences. The equation would be:
600= 3S + 2L
2L= 600-3S
L= 300-1.5S

The area equation would be:
A= S*L
A= S* (300-1.5S)
A= 300S- 1.5S^2

The maximum area equation would be:
A'= 300- 3S=0
3S= 300
S=100

The long side would be
L= 300-1.5S
L= 300- 1.5(100)= 150

The maximum area would be:
A= S*L= 100*150= 150000 ft^2