planes and pyramids

Question

design a single open box so that packages of size A completely fill your box, packages of size B completely fill your box, packages of size C completely fill your box, and packages of size D completely fill your box. 

Answer 1

Tiana,

 

This is just a least common multiple problem.  Assuming that each "box" is a polyhedron with 6 rectangular faces, we can stack them easily and tightly in a bigger box.

 

Each of those smaller boxes A, B, C, and has unknown dimensions in length, width and height: L_A, W_A,H_A for box A and similar L,W,H with subscripts B,C and D for the other boxes.

 

Then certainly the big box with dimensions L, W, & H would work:

where:

L = (L_A)(L_B)(L_C)(L_D)

W= (W_A)(W_B)(W_C)(W_D)

H = (H_A)(H_B)(H_C)(H_D)

 

A smaller box could also work if common factors existed across A, B, C & D.  For example, if you factored L_A, L_B, L_C & L_D separately and found that 2 was a factor in each, you could cut the dimension L of the big box in half and the smaller boxes would still fit as long as all L's go in the same direction, all w's in their same direction and all h's in their same direction.  But the problem states no specific dimensions, so we go with biggest box calculated above.