Answer :
Here is your answer
[tex]\bold{107.62} inches [/tex]
Since the door will be in shape of a rectangle where,
length= 80 inches
breadth= 72 inches
So,
widht of largest mattress that can fit diagonally is diagonal of the door.
i.e.
[tex]d= \sqrt{{l}^{2}+{b}^{2}}[/tex]
[tex]d= \sqrt{{80}^{2}+{72}^{2}}[/tex]
[tex]d= \sqrt{6400+5184}[/tex]
[tex]d= \sqrt{11584}[/tex]
[tex]d= 107.62[/tex]
HOPE IT IS USEFUL