.
From the context, the boxes are open.
The height of the box is just given: it is 6 inches and is not the subject of varying.
The perimeter of the box' base is given, too, as 60 inches.
The only dimensions to maximize the volume are the length and the width of the box, which must maximize the area of the base.
It is well known fact, that the base, under such conditions, must be a square with the side of 1/4 of the base perimeter.
Thus the base is a square with the side length of 60/4 = 15 inches.
Thus the original carton must be a square with the side length of (15+2*6) = 27 inches. ANSWER
Solved.
------------------
See the lesson
- A rectangle with a given perimeter which has the maximal area is a square
in this site.
/\/\/\/\/\/\/\/\/
Just a reminder for visitors (!)
When I solve a problem for you, I expect to get your "THANKS" back.
It is the simple and standard way to let me know that you are an educated person.
Otherwise, I will think that I work for idiots.