This problem came to me in a conversation with a friend this past weekend.
Airlines restrict the size of baggage by limiting the total linear dimensions of your bag. Specifically, the Width + Length + Height of your bag cannot exceed 60 inches. (For US airlines the restriction is actually 62″, but for simplicity we will use 60″ in this problem. Also, we will ignore the weight constraint airlines also place on bags, about 50 pounds per bag).
The Problem:
Suppose I went to Alaska salmon fishing and caught a bunch of fish. I then had the fish filleted, smoked, and packed. Note the packages are rather small and can be stacked very tightly together. Now I want to bring as much salmon home with me as I can. I have one bag to pack all the salmon into. What size bag will maximize the amount of salmon that I can bring home? You can design any type of rectangular bag that you want. For instance, the bag could have a width of 10″, a length of 42″, and a height of 8″. Remember the Width + Length + Height cannot exceed 60″.
I would expect that a middle school student should be able to get the correct answer to this problem with some time. Further, they should be able to heuristically support their work. I would expect a high school student studying Pre-Calc to get his problem correct (should not take a long time), but also be able to prove that their solution is correct.
Feel free to give my your answers below. I will post a detailed solution to this problem next week.
