 # Problem-Solving - Estimation and Orders of Magnitude

Author(s): Developing a good sense of how to use your physical intuition for solving problems involves learning the technique of deciding which approach to solving a problem tends to work more "easily" than other possible approaches. In some cases, detailed mathematical analysis is necessary, but, for many questions, an amazing degree of accuracy can be obtained by a series of "educated guesses" for which the correct order of magnitude for each quantity involved and simple calculations are all is all that is necessary to get a surprisingly accurate answer. The great nuclear physics Enrico Fermi is acknowledged as one of the great masters of this technique and used it to find the answers to what would seem to be almost impossible questions to answer. Hence the kind of problems that can be attacked in this way are termed Fermi problems. Here are some examples:

Example 1:
What thickness of rubber tread is worn off of the tire of a typical automobile as it travels one mile?

Solution:
We could answer this question if we knew the following information:

1. The thickness of tread on a new tire.
2. The average distance of driving before a tire typically needs to be replaced.
With this information we can calculate the amount of wear per unit distance driven and then multiply by a distance of 1 mile to get the tread wear. The utility of Fermi problem approaches is that, generally speaking, your knowledge of the world (physical intuition) will quantitatively not usually be off by more than an order of magnitude for each quantity you need to know. Since some estimates are too big and some too small compared to the true number, the average tends to be about right. Let's consider the assumptions needed here:
1. New tire tread thicknesses are about 1 inch. They may actually be 2 inches or 0.5 in, but an order of magnitude accuracy is "good enough". Certainly 0.1 inch is too small and 10 inches is too thick.

2. Typically tires will last about 40,000 miles. Again, the answer for some tires might be 30,000 miles and longer-wearing tires might be 60,000 miles, but you would find it hard to afford a car if the tires needed to be changed every 4,000 miles and very few people would ever need to buy tires if the answer was 400,000 miles.
Next, let's translate units and make the calculation.

 @
 é ê ë tread loss 1 tire ù ú û é ê ë 1 tire distance driven ù ú û
 @
 é ê ë 1 in. 40,000 mi. ù ú û
 @
 2.5×10-5 in./mi.

Although our answer may be off by a factor of 2 or 3, it is almost certainly not off by a factor of 10. For some problems this is more than adequate.

Example 2:
Fans of a college football team are excited after their team wins the game. They rush onto the field. How many people can fit onto the field of 100 yards by 50 yards?

Solution:
Start by assuming that each person needs an area of about 18 inches by 12 inches if they are packed as closely as possible. Then, we would have

[(100 yds)(3 ft/yd)(12 in/ft)/(18 in)][(50 yds)(3 ft/yd)(12 in/ft)/(12 in)] = 30,000

So 30,000 people can fit onto the field itself.

Sometimes you can use the Fermi method to get some amazingly accurate results from your everyday knowledge. For example, suppose you are asked to state the radius of the earth and you do not have that quantity handily available. Think about your (or others) travel experiences:

• If you fly across the United States, how many time zones do you cross? Answer: 3.
• What is the average distance across the US? Answer: about 3000 miles.
• So, on average, there are about 1000 miles of distance traveled per time zone.
• There must be 24 time zones around the earth in all since there are 24 hours in the day so the earth circumference must be about 24,000 miles.
• The circumference of a circle is 2pr where r is the radius so

r = (24,000 mi)/(2p) = 3820 miles = 6.2 x 106 m

The actual answer is 6.37 x 106 m so you can see that the Fermi estimate gets remarkably close. Although the breadth of the Internet and smart search engines have made it possible to quickly find incredible amounts of information, it is still useful to keep in mind some orders of magnitude for how big or small things are for just the kind of quick Fermi problem calculations we might use to decide how to attack particular problems. Here are a few quantities that might prove useful along with the Web links to where the information was derived. You may wish to explore these links at your leisure, but it is not necessary to do so. These links were valid as of Jun 1, 2001.  [Editor's note: Links checked Nov 8, 2004.]

World population (1999): 6.032 x 109 (estimated: 12/20/99) US Census Bureau
US population (1999): 2.74 x 108 (estimated: 12/20/99) US Census Bureau
World Population (1650): 5 x 108 Brockport HS Science Dept.
Power of a hurricane 2 x 1013 W Brockport HS Science Dept.
Height of Mount Everest 8,850 m Infoplease Everest Almanac
Length of a typical virus 10-8 m Tulane University

Problems:

1. How long would it take for you to walk from Los Angeles to Philadelphia?

2. How many dice would it take to fill a typical football stadium?

3. Each binary digit on a computer is called a bit. A series of eight bits forms a byte. A CD-ROM holds 650 megabytes of information. Estimate the number of typical books that can be stored on a CD-ROM.

Larry Gladney and Dennis DeTurck, "Problem-Solving - Estimation and Orders of Magnitude," Convergence (November 2004)

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