Batty About Buses (a superhero Fermi question)

Ok so the Joker has planted a bomb on one city bus and Batman must track it down before the end of rush hour. How many buses could it possibly be hidden on?

We might have some idea of how many blocks we would have to walk to catch a bus in a dense city like Gotham and how long we would have to wait until the bus would come. So then if we can guess how big Gotham city is we could make an estimated guess at how many buses it would need.

So how large an area is Gotham city? 1km^2? No idea? Ok, well let's say Gotham city is about the same size as NYC. So what's the size of NYC? I have no clue.

So what do we do? We compare to the size of something we know. I said that the area of the earth is 5*10^14 m^2. And how many NYC fit on earth? Ok, bad question but what's the size of the USA? Well how many USAs would fit in the world?

1? Bad guess must be bigger. 10? No still bigger. 100? Ya that's a good guess. But seems a little big.

So we take the geometric mean of 10 and 100 which is 10^((1+2)/2) = 30.

Ok so if the area of earth is 5*10^14 m^2 and 30 USAs fit on earth than the area of the USA is about... well 5/3 is about 1 ... come on! Remember we're estimating. So we say, ``who cares?'' Ok so 5/3=1 and 10^14 / 10 means we subtract the exponents 14-1=13 so (5/3) * 10^(14-1) m^2 = 10^13 m^2. Check out the "actual"' answer. So how many New Yorks fit in the USA? I don't know. But I do know how many states do. Assume all the states are about the same size and say there's a nice round number of them: 50 (if there weren't a nice round number we would round it. Like say 48 to 50). So the average area of each state is 10^13 / 50 = 2*10^11 m^2. So we estimate that New York state has an area of 2 * 10^11 m^2 and the official area of NY is 1.4*10^11 m^2. Not bad.

And finally to estimate the area of NYC: how many NYCs fit in NY state? 1? nah. 100? Maybe? 1000? Somewhere between 100 and 1000. So the geometric mean is 300. Then the area of NYC is about 2*10^11 m^2 / 300. 11-2=9 and 2/3 is about... oh I don't know 1. So then the area of NYC is 10^9 m^2. Actual area of NYC (drumroll please): 1.2144 * 10^9 m^2. Wow pretty damn close.

And Gotham is about the same size as New York. Ok now. How many buses? Let's guess how many roads are in Gotham. First let's say Gotham City is square. Then it has a width or length of the square root of the area. Remember from the last post that we square root by halving the exponent: =10^{3/2} km = 10 * 3km =30km. Ok and how many blocks are in 1km? 1? no. 100? no. 10? ya that sounds about right. 10 blocks for every 1km. That's like 20 city blocks in a mile. So if Gotham is 30km by 30km and there's 10 blocks per km then Gotham is 300 blocks by 300 blocks. The roads make grids around the blocks so there are 600 roads in Gotham city.

How many roads have buses on them? Every road? No way. Every 10 blocks? Too big. Geometric mean = 3. We guess that in a densely populated city like Gotham or New York there is a bus route every 3 blocks. So that means that of the 600 streets 200 have bus routes on them which each run for about 30 km (about the length of the city).

And how often do they come? Never fast enough, right? I don't know every 10 minutes seems fast but every 30 minutes seems slow. Let's guess every 20 minutes.

And how fast do they drive? Not 1km/hr that's slower than you walk (10 blocks in 1 hr? Sloooow) but 100km/hr which is like freeway speed. So Geometric mean = 30km/hr.

Ok so if the buses go 30km/hr and come every 20minutes then we can guess the distance between buses: 30 km / hr times (1 / 3 )hr = 1km.

We've got it!!! On each route there must be one bus every km over a 30 km distance and there are 200 routes. (1bus/km) times (30km/route) times (200routes) = 6000 buses in Gotham City.

That took a little while but the math was easy and we could have scribbled it on a napkin or maybe done it in our head. But how did we do?

Well... (and remember as long as we're within ten times bigger or ten times smaller we're pretty happy) according to the wikipedia article on the MTA fleet there are 6,251 buses in fixed route service. The article on the NYC transit buses says about 4500 buses. Either way, 6000 is a pretty good guess.