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

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.