Question

What is the greatest perimeter of a rectangle With an área of 39 square feet

Answer 1

x=lentht
y=width
area=xy

xy=39 ⇒x=39/y

Perimeter of a rectangle=2x+2y

P(x,y)=2x+2y
P(y)=2(39/y)+2y
P(y)=78/y+2y
P(y)=(78+2y²)/y

Therefore ;P(y) will be gratest if y⇒0

We calculate the next limit.

lim y⇒0 (78+2y²)/y=lim y⇒0 (78/y+2y)=∞

For exmple:
y=0.1 ft
then;; x=39/ 0.1=390 ft

P(2*390 ft+2*0.1 ft)=780.2

y=0.001 ft
Then; x=39/0.001=39000 ft

P(2*39000 ft+0.001 ft)=78000,002 ft.


Answer: the greatest perimeter of a rectangle with an area of 39 ft²⇒∞

Answer 2

There is no greatest perimeter.

The shortest perimeter for a fixed area is when you use
the area to make a circle.

The next shortest perimeter for a fixed area is when you
use the area to make a square.

But the longer and skinnier you make it, the longer the perimeter
becomes, without any limit.

Examples:

circle, radius = 3.523 . . area = 39 . perimeter = 22.14
√39 by √39 . . . . . area = 39 . . . . . perimeter = 24.98
6 by 6.5 . . . . . . . area = 39 . . . . . perimeter = 25

5 by 7.8 .. . . . . . . area = 39 . . . . . perimeter = 25.6
4 by 9.75 . .. . . . . area = 39 . . . . . perimeter = 27.5
3 by 13 . . . . . . . . area = 39 . . . . . perimeter = 32

2 by 19.5 . .. . . . . area = 39 . . . . . perimeter = 43
1 by 39 . . . . . . . . area = 39 . . . . . perimeter = 80
0.1 by 390 . . . . . area = 39 . . . . . perimeter = 780.2

0.01 by 3,900 . . area = 39 . . . . . perimeter = 7,800.02