A box without a top has dimensions z inches by y inches by z inches. The cost of the material for the base of the box is 4 cents per square inch.

A box without a top has dimensions z inches by y inches by z inches. The cost of the material for the base of the box is 4 cents per square inch. The cost of the material for the sides of the box is 1 cent per square inch. If the total budget is 48 cents, what are the dimensions of the box with the largest volume that can be produced? You may assume without justification that only an absolute max exists and the dimensions that occur at this max are all strictly positive.