(10)

Step 1:

Perimeter of the three pens is ft

Let each pen has length ft and breadth ft.

Since there are three pens total length is .

Total width is .

ft.

. \ \

Area of the total fencing is .

Substitute .

.

The area of the rectangular field is always positive.

and .

and .

is  positive on the interval .

Step 2:

Find the dimensions of the rectangular field, that will enclose the maximum area.

Apply derivative on each side with respect to .

Find the critical numbers by equating .

ft.

Substitute in .

 ft.

Step 3:

The maximum value of occurs at either at critical number or at end point of the interval .

Substitute in .

.

Substitute in .

.

Substitute in .

sq ft. \ \

The maximum area of a rectangular fencing is sq ft. \ \

The dimensions of the rectangular field are ft and ft. \ \

Solution:

The dimensions of the rectangular field are ft and ft.

The maximum area of a rectangular fencing is sq ft.