UPS defines the size of a box to be equal to its length plus twice its width plus twice its height: size = length + 2 * width + 2 * height In order to avoid being charged freight charges, the size of a package has to be less than 165" and the length has to be less than 108". You work for a wine producer who ships wine bottles in rectangular boxes that are 13" x 4" x 4". The individual boxes are packaged into larger rectangular boxes for shipping. In order to go through the company's production line, the length of the boxes has to be 13". The company wants to minimize its shipping costs so they need to send as few of the large boxes as possible. What size box will hold the maximum number of the smaller boxes?

If there were no restrictions on the size of the shipping container, dimensions that contribute to the sum (length + 2·width + 2·height) would be equally distributed between the three contributors to the sum. That is, each member of the sum would be about (165 inches)/3 = 55 inches.

The closest we can get to that using a multiple of 13 inches is 52 inches. If the length is 52 inches, then the sides of the square box can be (165-52)/4 = 28.25 inches. The closest dimension that is a multiple of 4 inches is 28 inches.

The optimum shipping container will be 52 inches long by 28 inches square. It will hold (52/13)(28/4)^2 = 196 bottles of wine.