Last week, James Cleveland (@jacehan) shared this:

It is weird. You would think the size limit would be volume, not combined length, right?

The first question that came to my mind was “What are the dimensions of the bag with the greatest volume?”

A “cubey” bag with a length and width of 21 inches and height of 20 inches would have a volume of 8820 cubic inches, or 5.1 cubic feet. The airlines are banking on your bag looking more like the one pictured above. The dimensions are not shown, so let’s assume the golden ratio is at play here:

w + l + h = 62

w + w(1.618) + w(1.618²) = 62

w(1 − 1.618³)/(1 – 1.618) = 62

w = 11.84

w = 12 in, l = 19 in, h = 31 in

A “golden” bag would have a volume of 7068 cubic inches, or 4.1 cubic feet. If passengers were able to check a “cubey” bag, they’d be able to pack about 25% more. Of course, the airlines would still get ’em with the weight limit.

I guess it does make sense to express the size limit in inches rather than inches cubed. After all, a bag with a length and width of 1 inch and height of 7068 inches would also have a volume of 7068 cubic inches.

Math teachers have seen this type of problem before, but never like this. We’ve seen farmers with x feet of fencing faced with the challenge of enclosing the largest possible pig pen. In later grades, we insist that the farmer use the exterior of the barn as one side. Length is given and area is maximized. This can be reversed. That is, given the size of the pen, our farmer must use the least amount of fencing.

We’ve seen problems in which surface area is given and volume is maximized (like the popcorn box problem or the rolling paper into cylinders thing). Again, this can be reversed. Timon’s Piccini’s pop box design task is in this family.

The checked baggage problem, on the other hand, jumps a dimension. We’ve never seen problems in which length is given and volume is maximized. I wonder if this opens up some interesting possibilities.