Skip links

Small-Scale Calculations

I’ve been packing for a move, which means a lot of assembling boxes from the movers and filling them. There’s some fun in the repetitive tasks and there’s also a lot of time to think, which has in the past led me down some strange paths.

The book boxes are 17 inches by 12-1/2 by 12-1/2. They’re marked as 1.5 cubic feet, and the dimensions give me 1.53 c.f., so close enough. The platonic ideal of packing a book box is that you perfectly fill the entire interior with books, so that there is no leftover space and the box, taped, is exactly the advertised dimensions. The density of paper varies, but a commonly-used middle value is 75 pounds per cubic foot. So that standard box, perfectly filled with paper, would weigh 113 pounds.

It is impossible to perfectly fill a box with books, as anyone who has ever packed books knows. Books are made to many different sizes – sometimes, for some inexplicable reason, with very small variations – and their size changes a bit with use. Paperbacks change more than hardcovers, but all books pick up some moisture and oil from our hands, gain some creases, and so on. Most importantly, there is no one size of box that will be a whole-number multiplier of all of the standard sizes of books, so there is inevitably some space left over. In short, if the books were rectangular prisms, they wouldn’t perfectly fit in the box, and the books aren’t perfect rectangular prisms. The amount of empty space from varies box to box, but I’d put it in the 10 to 15 percent range. So the boxes actually weigh more like 90 to 100 pounds. That’s why book boxes are the smallest size boxes that the movers use: that’s heavy enough. The “linen” boxes that we’re using for less-dense objects are nominally 4 cubic feet in volume, and asking the movers to deal with a box box that weighs 300 pounds is a bit much.

I don’t know how that book box size was found – possibly by trial and error – but there’s an interesting possibility. At the end of the 1800s and beginning of the 1900s, there was a trend towards using time-and-motion studies to rationalize industrial productivity. This idea – most famously applied by Frederick Winslow Taylor and by Frank and Lillian Gilbreth – was to break down any physical labor task into its component steps, time each step, and so set an expected time for the entire task. There are a bunch of problems with the concept, from the difference between different people’s abilities, to the fact that there may not be a single ideal way to perform a task, to the fact that the speed at which a laborer performs a task will vary through the day. Part of the supposedly rational approach was to look at the ideal load a laborer could carry by hand, or move with a shovel, or any one of a hundred other possibilities. I would not be at all surprised to find that a roughly 100-pound box was first proposed by Taylor or one of his followers.