# Small angles

In Chapter 4 of Misbehaving, Richard Thaler writes:

Here is a test to see if you are a good intuitive Pythagorean thinker.

I mentioned this test in my previous post. Here’s Thaler’s formulation of it, abridged by me:

Consider two pieces of railroad track, each one mile long, laid end to end… The tracks are nailed down at their end points but simply meet in the middle. Now, suppose it gets hot and the railroad tracks expand, each by one inch. …the tracks can only expand by rising like a drawbridge.

They are also extremely sturdy (and light, I would add) so they don’t droop or curve but remain perfectly straight. It would be interesting to see what would happen if they started curving instead of rising but that’s not the point of the exercise. Thaler’s question follows:

Consider just one side of the track. We have a right triangle with a base of one mile, a hypotenuse of one mile plus one inch. What is the altitude? In other words, by how much does the track rise above the ground?

It’s remarkable how spectacularly wrong the average answer was: 2 inches instead of around 30 feet, two orders of magnitude off. However, this experiment would have more relevance to behavioral economics if the respondents had “skin in the game,” to use one of N. N. Taleb’s favorite expressions, – if, say, they were fined for getting the answer wrong by more than 50%. In addition, the image of a whole mile of impeccably rigid, feather-light track does not strongly appeal to our intuition, Pythagorean or otherwise. I’d consider using a model railroad instead.

I estimated the answer while showering, taking shortcuts whenever I could. At first, I thought of using one mile as the base unit but dealing with small fractions would be a pain. If L is the length of a mile in inches and H is the altitude we’re looking for (also in inches), then

$latex H=\sqrt{(L+1)^2-L^2}=\sqrt{2L+1}$.

One inch is negligible compared with 2L so the question is, what’s the square root of twice the number of inches in a mile?

My brain is strictly metric; it also hates division by non-integers. One inch (that much I’ve always known, of course) is 2.5 cm (2.54 more precisely but the third digit only adds 2% and I didn’t remember it anyway) or 25 mm. I know the mile is somewhere between 150 * 10,000 and 175 * 10,000 millimeters, 150 and 175 being multiples of 25. This gives me 6 to 7 inches multiplied by 10,000. Doubling leaves me with 12-14 by 10,000. This is rather convenient for square root extraction: we’re in the 300-400 inch range, probably not much less than 350. Conveniently, one foot is about 30 cm, roughly 12 inches: 360 is 30 dozen, so my guess was 30 feet. Bingo.

I’ve thought up a more entertaining example. The Earth’s radius is about 6,400 km. One hundred meters is to this radius roughly as one inch to a mile. Suppose we have drilled a strictly vertical well – I’m thinking of an oil or gas well but any straight borehole will do – 100 m (109 yards) deep. (Shallow by the oil industry’s standards.) Then, we start drilling a horizontal section strictly perpendicular to the vertical wellbore. Suppose our omnipotent equipment lets us drill horizontally until the drill bit pops out of the earth because of the surface curvature. How long would this horizontal section be? It turns out 36 km (22 miles) long, three times the longest horizontal reach in the industry (12 km at Sakhalin-1, operated by Exxon). By the way, one can use 6,000 km or 7,000 km as a rough guess of the Earth’s radius and get pretty much the same result. A hundred yards deep and twenty miles long.

A similar question, probably familiar to all from school-level physics or astronomy, is how far a sailor can see from a crow’s nest, let’s say, only 30 feet above the water level in good weather. The answer is 6.7 miles. However, tripling the height of the lookout point to 90 feet (the crow’s nest on Titanic) would only bring the visibility up into the 11-12 miles range: the square root is a relatively slow grower.

Another way to look at this slightly baffling phenomenon is to recall that for small angles, sin x behaves like x while (1 – cos x) behaves like x²/2. The angles between the hypotenuse and the longer cathetus in Thaler’s problem and in my horizontal well example are a little larger than 0.3°.

What relevance does it have to economic rationality? Not much, I’m afraid. Humans make estimation blunders in every thinkable field, but it’s not a Nobel-grade revelation.