Let w be the original width of the sheet, in centimeters.

Then the original length was (w + 40) cm.


After cutting squares at the corners, folding and soldering, the base of the box has 

dimensions (w-2*15) = (w-30) cm  and  (w+40-2*15) = (w+10) cm.



Therefore, the volume of the open box is  (w-30)*(w+10)*15 cm^3.



It gives you an equation for the volume

    (w-30)*(w+10)*15 = 67500  cm^3.


Cancel by the factor 15 both sides

     (w-30)*(w+10) = 4500 


You can transform this equation to the standard quadratic equation form and solve it using the quadratic formula.


You can also notice that the only decomposition of the number 4500 into the product of two numbers 

with the difference of 40 is  4500 = 50*90,

so  you can get the solution MENTALLY  w-30 = 50,  w = 50+30 = 80.


ANSWER.  The dimensions of the original rectangular sheet were  80 cm (width) and 80+40 = 120 cm (length).