Your job is to design a closed rectangular box of minimum cost. Suppose the box is to be of volume cubic cm; and the cost of material for the front and back sides is b dollars per square cm, c dollars per square cm for the left and right two sides, and d dollars per square cm for the top and bottom sides.

1) Do the mathematics to find the dimensions of the box of minimum cost.
2) Explain the ramifications of various differences in the costs of materials, i.e. what happens to the dimensions of the box. You might want to deal with certain cases:
i) b = c = d,
ii) b = c = 2d,
iii) b = 2c = 3d.