About how many recordings are played by the radio stations in Ontario in a day?

Rich estimation problems like this one are also known as Fermi problems. They are named after Enrico Fermi (1901-1954). He was a leading research scientist, who won the Nobel Prize for physics in 1938 and spent the last part of his career as a professor at the University of Chicago. He liked to show his students that they had the knowledge to answer seemingly impossible questions that he posed.

The importance of Fermi problems is to illustrate the difference between guessing and estimation. Although guessing may produce a reasonable answer to a problem, you do not now how much confidence to place in the answer. When estimating the answers to Fermi problems, you will need to make some assumptions. If the estimated answer seems to be unreasonable, go back and check your assumptions.

1. What information are you given?
2. What are you asked to find?
3. Do you need an exact or an approximate answer?

The problem is a volume problem, in which you need to estimate how many small objects are needed to fill a large object. The number of small objects, n, can be found using:

n = (large volume) ÷ (small volume)

You can use your research skills to find that the inside of a telephone booth approximates a square-based prism. The side length of the base is about 0.9 m, and the height is about 1.9 m.

Assume the peanuts in a shell approximate a cylinder with a diameter of about 1.5 cm and a height of about 4 cm.

The volume of a telephone booth, in cubic centimetres, is about 90 x 90 x 190, or about 1 540 000 cm³.

So, n = 1 540 000 ÷ 7

= 220 000

About 2220 000 peanuts in the shell would be needed to fill a telephone booth.

Does the answer seem reasonable?

Is there a way to improve the estimate?

In summary, to solve Rich Estimation Problems, follow these steps:

1. Locate the information you need.
2. Decide what assumption(s) to make.
3. Estimate the solution to the problem.
4. Check that your estimate is reasonable.