What the is greatest possible volume of the box?

Question

The box will not have a top and it will be made by cutting out corners of equal size from a 28 inch by 15 inch sheet of cardboard and folding up the sides.

Answer 1

If x is the length of the cut (which becomes the height of the box) we have:

 

V= (28-2x)(15-2x)x

V= 420x -86x^2+4x^3

dV/dx=  420 -172x +12x^2

dV/dx= 105-43x +3x^2

Set equal to 0 and use quadratic formula to solve for x.

x=3.1218

So the volume is 21.7564*8.7564*3.1218= 594.73 or 595 cu in