Question

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.

Answer 1

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