SOLUTION: Find the dimensions of the box described. The length is 5 inches more than the width. The width is 4 inches more than the height. The volume is 416 cubic inches. Find the length,

Let w be the width.


Then the length is (w+5) and the height is (w-4).


The equation for the volume is


    w*(w+5)*(w-4) = 416.


Notice that  416 = 8*13*4 - I decomposed the number 416 into the product of 3 factors.


It can be done by different ways, but I took the care that the second factor is 5 more than the first and the third is 4 less than the first -

exactly as the numbers w, (w+5) and (w-4) in the left side of equation (1).


It gives me a confidence that I correctly guessed the solution.



The plot below shows that the guessed solution w = 8 is UNIQUE.



    


    Plot y = x*(x+5)*(x-4) (red) and y = 416 (green)


ANSWER.  The width is 8 inches;  the length is  8+5 = 13 inches;  the height is  8-4 = 4 inches.