by Kevin B. English
The following is the solution to the luggage problem that I proposed a week or two ago.
To refresh your memories here is the original problem. Click here to read the full post of the original problem.
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″.
The Answer:
Width = 20″
Length = 20″
Height = 20″
Or, in words, the rectangular box design that gives the greatest volume is a cube (all sides of equal length).
Middle School Student – Justification
I would expect a middle school student to justify their answer through a series of examples. For example,
Trial # | Width | Length | Height | Volume (= width X length X height)
1 1 1 58 V = 1 X 1 X 58 = 58 cubic inches
2 10 10 40 V = 10 X 10 X 40 = 4000 cubic inches
3 15 15 30 V = 15 X 15 X 30 = 6750 cubic inches
4 20 20 20 V = 20 X 20 X 20 = 8000 cubic inches
5 19 21 20 V = 19 X 21 X 20 = 7980 cubic inches
From this table, and this table is the least amount of work I would expect as a teacher, I would expect the student to venture a conclusion that 20 inches by 20 inches by 20 inches gives the greatest volume.
High School Student – Justification
I would expect a high school student who has completed Pre-Calculus to get the correct solution and provide a proof or illustration of why 20 inches by 20 inches by 20 inches is correct and there are no other solutions. Note that in the Middle School solution we found the answer by trial but we could not guarantee that we have the correct answer. There are an infinite number of possibilities for the rectangular box design and we only checked 5 possibilities. For example, what about the possibility of 19.8 by 20.2 by 20? (This gives a volume of 7999.2 cubic inches.)
Here is what I would expect of a high school student. Continue reading →