# 1. Some Background on Estimation

(This refers to Homework 1.)

Physics is a quantitative science and so any appreciation of its contribution to the modern world view obliges us to do some mathematics. We will use some arithmetic, algebra, geometry, and "scientific notation". Don't panic, you'll be surprised at how little is necessary.

The first thing we are going to talk about is estimation of numbers. The ability to estimate the size or probability of various phenomena is useful in many realms of activity beyond science. It is worth developing for the sake of having a better grasp of economic and political decision making in the public and personal spheres as well.

We begin with some estimations that, to the uninitiated, seem totally impossible to accomplish with any confidence. You'll be surprised!

## 1.1. How many piano tuners are there in the greater Boston area?

Solution:

The class first approximated the total population to be 600,000 for Boston and about 2 million for the area that includes suburbs. That is 2×10^{6} people using scientific notation.

Then the number of households was determined by assuming from 2 to 4 people per household, yielding 2×10^{6} ÷ 2 = 1.×10^{6} down to 2×10^{6} ÷ 4 = 5.×10^{5 }households.

Next it was assumed that 10% of households have pianos, leaving 5×10^{4} to 1×10^{5} pianos to be tuned.

Students estimated that an average piano gets tuned once per year. They think it takes about 1 hour for a tuner to tune along with 1 hour traveling time. Then one tuner working about 8 hours a day will tune 4 pianos per day. If the tuner works about 200 days per year she can tune 800 pianos, or roughly 103 per year. That is 103 pianos/year/tuner.

So 5×10^{4} to 1×10^{5} pianos tuned per year ÷ 10^{3} pianos/year/tuner = 50 to 100 piano tuners.

The number of entries in the Yellow Pages is 62. So we are definitely in the right order of magnitude.