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Fermi Problems

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Fermi was famous for doing "back of the envelope" calculations in which the information given seems incomplete. An example of solving a Fermi Problem.

The problem we will solve is: How many piano tuners are in New York City? How might one figure out such a thing? Surely the number of piano tuners in some way depends on the number of pianos. The number of pianos must connect in some way to the number of people in the area.

Approximately how many people are in the NYC? 4,000,000

Does every individual own a piano? No

Would it be reasonable to assert that individuals don't tend to own pianos; households do? Yes

About how many households are there in NYC? Perhaps 1,000,000

Does every household own a piano? No. Perhaps one out of every five does. That
would mean there are about 200,000 pianos in NYC.

Do all of these need tuning? No. Perhaps 1/2 of the pianos in NYC are electronic and
don't require tuning. Thus, there are about 100,000 pianos in NYC that need tuning.

What about the pianos in the Conservatory of Music?
Probably negligible compared to the 100,000 pianos owned privately.

How many pianos are tuned every year? Some people never get around to tuning their piano; some people tune their piano every month. If we assume that "on the average" every piano gets tuned every two years, then there are 50,000 "piano tunings" every year.

How many piano tunings can one piano tuner do? Let's assume that the average piano tuner can tune three pianos a day. Also assume that there are 250 working days per year. That means that every tuner can tune about 750 pianos per year.

Finally, then how many piano tuners are needed in NYC? The number of tuners
needed is about 50,000/750 = 67. Assuming we have just enough tuners to satisfy the demand, there are therefore 67 piano tuners in NYC.

Using different assumptions for various factors, it is unlikely that you can justify an
answer greater than a factor of 10 or smaller than a factor of 10 from the number we just obtained; that is to say, there are probably not more than 670 tuners and surely no less than 7. Thus the answer obtained is probably good to within an "order of magnitude."

Solve the following problem based on what you learnt from the above example. Explain all the steps in your reasoning.

How many hairs are on your head?

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Solution Summary

The solution contains answers to a couple of Fermi Problems which include the hair problem and also the number of pianos in the New York city.

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Please see the attached file for the answers.

Enrico Fermi (1901-1954) is an Italian physicist best known for his contributions to nuclear physics and the development of quantum theory. In addition to his contribution to theory, he is also noted as an experimentalist.

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