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Question

Answer 1

(1)

Step 1:

The total material is required to fence a rectangular field is ft.

Let the rectangular field has length ft and breadth ft.

The perimeter of the rectangular field is .

.

The area of the rectangular field is .

Substitute in .

.

The area of the rectangular field is always positive.

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 .

.

To find the critical numbers by equating .

ft.

Comments

Contd...

Step 3:

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

Substitute in .

.

Substitute in .

.

Substitute in .

.

The area maximum at length of rectangular field is ft.

Substitute in .

ft.

The dimensions of the rectangular field are ft and ft.

Solution:

The dimensions of the rectangular field are ft and ft.

 

Answer 3

(2)

Step 1:

The rectangular field area is square ft.

Let the rectangular field has length ft and breadth ft.

The area of the rectangular field is .

.

The perimeter of the rectangular field is .

Substitute in .

.

The perimeter of the rectangular field is always positive.

Perimeter is positive on the interval .

Step 2:

Find the dimensions of the rectangular field, that will require least amount of fencing.

Apply derivative on each side with respect to .

.

Find the critical numbers by equating .

ft.

Comments

Contd...

Step 3:

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

is negative for .

 is positive for .

Since is decreasing for all to the left of the critical number and increasing for all to the right, must give rise to an absolute minimum.

Substitute in .

.

The dimensions of the rectangular field are ft and ft.

Solution:

The dimensions of the rectangular field are ft and ft.