There is no greatest perimeter of a rectangle with an area of 39 square feet.
To see why, consider the length of 3 and width of 13. Area would be 39 and perimeter would be 3 + 3 + 13 + 13 = 32
Now consider length of 3/2 = 1.5 and 13·2 = 26. Then area is same but perimeter is 1.5 + 1.5 + 26 + 26 = 55.
Now we can just repeatedly half the length and double the width.
L = 3.0, W = 13
⇒ A = 39, P = 32.0
L = 1.5, W = 26
⇒ A = 39, P = 55.0
L = 0.75, W = 52
⇒ A = 39, P = 105.5
L = 0.375, W = 104
⇒ A = 39, P = 208.75
L = 0.1875, W = 208
⇒ A = 39, P = 416.375
L = 0.09375, W = 416
⇒ A = 39, P = 832.1875
L = 0.046875, W = 832
⇒ A = 39, P = 1664.09375
L = 0.0234375, W = 1664
⇒ A = 39, P = 3328.046875
L = 0.01171875, W = 3328
⇒ A = 39, P = 6656.0234375
L = 0.005859375, W = 6656
⇒ A = 39, P = 13312.01171875
Notice how P just kept growing. That's because width grow much faster than length shrinking, so P would grow endlessly.
Hope this helps.
