box volume problem

Question

 An open rectangular box can be formed by cutting identical squares off the four comers
of a rectangular piece of cardboard and folding up the-four sections that stick out.

 

For a particular piece of cardboard, the same volume results whether squares of size 1 or

squares of size 2 have been cut out.

 

Compute the resulting volume if squares of size 3 are
cut out.

Answer 1

Let L = length of cardboard and W = width

 

If size 1 squares are cut out, then volume of box = (L-2)(W-2)(1)

 

If size 2 squares are cut out, then volume of box = (L-4)(W-4)(2)

 

So, LW - 2L - 2W +4 = 2(LW - 4L - 4W + 16)

 

     LW - 2L - 2W + 4 = 2LW - 8L - 8W + 32

 

 Therefore, LW = 6L + 6W - 28

 

If size 3 squares are cut out, then volume = (L-6)(W-6)(3)

 

                                                              = 3(LW - 6L - 6W +36)

 

                                                              = 3(6L + 6W - 28 - 6L - 6W + 36)

 

                                                              = 3(8) = 24