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.

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I have wondered about the linear-measurements requirement. There’s a similar one at the (United States) post office regarding boxes. I suppose the bureaucratic thought behind it is that the average customer is going to be fairly well able to measure the length and width and height of a bundle and add those up, but asking people to find the volume of something is going to be epochally difficult. (It shouldn’t be, but then, watch a couple rounds of the Check Game pricing game on The Price Is Right — the only challenge in it is that the contestant has to subtract, and it’s so routinely a disaster they rarely play the game anymore.) Trusting that the proportions of things aren’t outside some reasonable bounds, then, you get more or less the volume restriction you really want without having to explain anything trickier than addition to people.

(It might also be so that airline personnel can veto a bag easily; showing that length plus width plus height exceeds some maximum bounds can be done in seconds, and doesn’t leave room for argument, while volume calculations … shouldn’t leave room for argument, but need more than a tape measure to prove.)

Yep, I think the ease/speed of customers/airline personnel to check/veto bags is what drives this requirement. Interesting, to me anyway, is that the airlines are less trusting that the proportions aren’t outside of those reasonable bounds when it comes to carry-on. Here, they state the maximum dimensions (9″ by 15.5″ by 21.5″) and that your-bag-must-fit-in-here-thing, not a tape measure, settles any arguments. You could ask students why.

Another thing that amused me in the picture was the metric precision. I imagined US airline personnel being given tape measures and their Canadian counterparts being given giant calipers.

I have problems with the in3 confusing kids. They’re not sure if the number is already cubed or not. A lot of them try to “simplify” 8 in3 to 512 in.

I think the linear total is probably just easier for more people to understand (and generally ignore, in my recent airline experience).