The fine art of approximation

It has been said that, during the first atomic bomb test, Enrico Fermi wanted a quick estimate of the energy of the blast. So, as the shock wave hit, he tossed a handful of paper scraps in to the air and watched how far they were carried. He estimated that the energy was about ten kilotons – remarkably close to the measured value of twenty.

Whether this actually happened or not, is a subject for historians to debate, but it makes for a good story nonetheless. And it serves to illustrate how a quick and dirty estimate can aid in decision making. In science classrooms around the world, these sorts of approximation problems are used to teach “science thinking” without getting bogged down in math. And, like all good science tools, it’s partly a matter of convenience and partly a matter of laziness. Some approximation exercises that I remember from my own schooling:

  • How fast would you have to stir your coffee in order to make it boil?
  • How many gas stations are there in the United States?
  • If everyone in China faced west and sneezed at the same time, how would the earth’s rotation change?
  • How fast would you have to drive a car through a hard rain in order to meet a wall of water?

With these types of problems, it isn’t the answer that is interesting but how one arrives at it. And once you get the hang of this you can get a surprising level of accuracy, particularly if you know which way to fudge the numbers. Of course, there are a few tricks to this kind of “Fermi estimate”. Trick number one, don’t care too much about what the answer actually is. If you’re attached to an outcome, you may subconsciously pick numbers that steer toward it.

Trick two, go fast the first time around and then fix it later. The first pass-through is just to get the process right. In subsequent estimations, you can try to get better numbers or to include things that you hadn’t previously thought of.

Trick three, round to the nearest whatever. Some numbers are easier to work with than others. You can work with powers of ten just by moving the decimal point. Computer engineers and programmers know the powers of two better than their own phone numbers. Once you get a feel for how numbers themselves work, then calculation becomes a snap. And because you don’t care about the end number, you can feel free to round off a bit.

Here’s an interesting example…

Global air-conditioning

One morning, I was driving up to a nowhere spot in central California to meet with a client. Radio coverage was essentially non-existent and so I ended up listening to someone on AM talk radio. This someone made the claim that if global warming is man-made, it’s probably from everyone running their air conditioners. Let’s look at this claim and construct a very simple model.

There are approximately four hundred million people in the US right now (and we’re going to ignore Alaska – they probably don’t do too much air conditioning). We’ll assume that each and every person has a five thousand square foot home, five thousand square foot office, and ninety thousand square feet representing their share of communal space (public buildings, malls, etc.). So every man, woman, and child has their very own air-conditioned area of one hundred-thousand square feet. Further, we shall assume that their ceilings are ten feet high, giving each person a million cubic feet of air-conditioned bliss. Four hundred trillion cubic feet in total (notice all of the powers of ten that I’m using).

How cold do they like it? Let’s further assume that every person in the US is currently trying to fight one hundred degree weather and cool their space down to seventy degrees. If air conditioners were one hundred percent efficient (spoiler: they are not) then we’d have to warm a like mass of air by thirty degrees. For our initial model, we’ll assume that air conditioners are only twenty percent efficient (probably they’re a bit better than this, but this is closer to the truth) and so we will have to warm five times that volume (five is almost as easy to use as two and ten).

So, in our hypothetical model, we’re warming up two quadrillion cubic feet of air by thirty degrees. That’s a lot of air. Let’s convert to cubic miles, for sake of readability:

(2,000,000,000,000,000) / (5280 x 5280 x 5280)

So about thirteen thousand cubic miles – a much more manageable number.

How much air is there in the continental United States? According to Wiki, there are about three million square miles of surface area. The atmosphere extends upward to about sixty miles, but most of the action takes place within three miles of the surface, so let’s just use that and approximate that there are about ten million cubic miles of air.

Divide the one in to the other, and the air-conditioned-warmed air represents only one tenth of a percent (note that I’m doing a lot of rounding here) of the volume of air in the United States. All of our air conditioning would warm that mass by three one-hundredths of a degree.

But (and this is a big “but”), our model assumes that all cooling and all air is evenly distributed all over the country by the same amount, everywhere. This simply isn’t true. Additionally, it isn’t true that every person has that much cooled volume. And finally, it isn’t true that every person in the country requires cooling by thirty degrees, all at once. Cities like Phoenix may require more cooling all in one spot; and places like Seattle may not require any. So we can see some ways to begin to refine our model.

I’m not going to argue that air conditioning causes or doesn’t cause warming. It may actually have a measurable (though tiny) effect in some places. The point of this exercise is to show both the power and the peril of making a casual model.

My homework assignment to you: play with this (either on paper or in your head)! See if you can think of ways to refine it. See if you can think of wrenches to throw in to the works. See if you can find some better numbers to use. Feel free to cheat and use the internet if you get lazy (but please give it a go, first).

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