help - derivatives-??

Question

Answer 1

(10)

Step 1:

Perimeter of the three pens is ft

Let each pen has length ft and breadth ft.

Observe the figure:

The total perimeter of the three pens is .

.

Area of the total fencing is .

Substitute .

.

The area of the rectangular field is always positive.

and .

and .

is  positive on the interval .

Answer 2

Contd...

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.

The dimensions of the rectangular field are ft and ft.

Comments

Contd...

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.

Solution:

The dimensions of the each pen are ft and ft.

The maximum area of a rectangular fencing is sq ft.