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

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

D.