The Basics of Fermi Estimates

Introduction

How many hours do all Americans combined spend on the Internet in one year?
How much area is covered by all of the interstate roads in the United States?
What volume of pen ink is used per day by all of the college students in the world?
How many days worth of calories would Superman burn if he ran a lap around the equator?

For questions like these, you usually don’t have any means of obtaining an exact answer. That’s where estimating comes in!

I got the idea to introduce these types of exercises to the LTC students from my senior seminar professors at Dickinson, who would assign us a new estimation problem every week for homework. At the beginning of class, each of us would reveal our final answers, and after a couple of weeks of practicing various types of estimations, we all started coming up with results that were within one order of magnitude of each other! I think that’s pretty impressive, considering we often approached the questions from different directions.

Working through estimation problems helps to improve your critical thinking skills, because often times you have to really think outside-of-the-box to figure out the most efficient method for tackling the question. This will inevitably stretch the way you approach solving problems in general.

As you work through more and more estimations, you’ll start to develop a better intuition about various quantities and create a mental bank for yourself with all sorts of useful pieces of information. Furthermore, the results you end up are usually the answers to interesting questions, so you’re bound to come away from these exercises with a slew of fun facts that you can share with (and use to impress) others.

Tips for getting started

1. Break down the problem

Even the most overwhelming and impossible-sounding questions can be divided into (relatively) simpler parts. Brainstorm each piece of information that you need in order to come out with the final product.

(e.g. If you’re estimating the amount of time it will take to climb a staircase to the moon, you’ll need to figure out: (1) a distance to the moon, (2) how many stairs that would translate to, and (3) the average time it takes to climb one step)

2. Look at the big picture

Don’t get too caught up in small details, but rather use a broader mindset when approaching the question.

(e.g. To estimate the volume of all of the buildings in the U.S., think about how each person has a space where they live and a space where they work, instead of trying to take into account each house, skyscraper, movie theater, tool shed, etc.)

3. Orders of magnitude are what matter most

If you get stuck on estimating a certain piece of information, think in terms of boundary values. Ask yourself, what is a reasonable assumption about the minimum and maximum value for this quantity? After you set an upper and lower limit, take the approximate geometric mean.

4. Pay close attention to units

Always carry your units throughout the entire problem, and be careful when you make conversions. Whenever possible, work in scientific notation and stick with the metric system.

It’s often tempting to just plug your question about a certain quantity into a search engine to see what the result is. However, you’d be surprised by how close you can get to the right answer if you take the time to reason out a plausible value.

After you come up with your own answer, it could be interesting to compare this with an accepted value (if there is one). Just don't peek too soon.

6. What does the answer mean?

Oftentimes the result will be a very, very large or very, very small number. Either way, it doesn’t mean much if it’s an unfathomable amount. Therefore, try to compare the answer you end up with to something you can actually picture or comprehend.

(e.g. If you estimate that the U.S. would save 3 x 1018 J/year if every household switched over to compact fluorescent bulbs, put the result in terms of the number of power plants needed to output that amount of energy per year.)

This week's estimation

If you paved a pathway to the moon, how long would it take (in years) to:
(a) walk it? (b) jog it? (c) sprint it?

Previous estimations

(Results obtained by Melia, William, Kathy, Kevin, and Samantha)

Friday 25 July 2013 - If we had a red HeNe laser with a Fabry-Pérot cavity the length of Long Island:

(A) What is the frequency spacing between two adjacent longitudinal modes in the cavity?

Result: Answers ranged from 600 to 800 Hz

(B) How many lasing modes would be present at a given time?

Result: Since the bandwidth for a HeNe laser is 1.5 GHz, this would result in 2 million modes!

Monday 22 July 2013 - If we covered all of the roofs of buildings on Stony Brook campus with solar cells, how much energy could we produce in 12 hours? [kWhr] (assuming constant, unobstructed sunlight during this period)

• Area of roofs on Stony Brook campus

• Intensity of the sun

• Efficiency of a solar panel

Result: Answers ranged between 2 x 104 to 6 x 106 kWh

What does this mean?

• Melia: 1 x 105 kWh would power an average American household for 5 years.

• William: 9 x 105 kWh could power 900 skyscrapers for a day.

• Kevin: 6 x 106 kWh is enough to keep 6,000 average American homes operational for a month.

• Kathy: 9 x 104 kWh can run a 100W lightbulb for a century.

• Samantha: 2 x 104 kWh is about equal to the energy of a lightning bolt.

Friday 12 July 2013 - What is the maximum (“best”) angular resolution of the human eye?
i.e. What is the smallest item you can see at a certain distance (based on the physical limitations of the eye)?

One method involves the equation for the angular spread of light through a pinhole, where 1.22 comes from the first zero of the Bessel function. (This actually corresponds to the boundary of the central bright spot, with respect to the optical axis, of the Airy diffraction pattern).

What does this mean: If the maximum angular resolution is 2 x 10-4 radians, this means that the smallest object that your eye can resolve from one mile away is about 1 foot high.

Tuesday 9 July 2013 - If we use a red HeNe laser and the Umbilic Torus as our aperture:

(A)What are the focal lengths of the lenses you would need to expand the beam enough to fill this aperture?

Result: Using a Keplerian beam expander with two lenses, f1/f2 pairs were 1mm/1m, 10mm/5m, 1mm/5m, 1x/2000x

(B) How far would we have to go to see the far-field diffraction pattern? [km]

Result: Using the Fresnel number, L >> answers ranged from 6 x 103 to 3 x 105 km

What does this mean? 105 km is the same order of magnitude as our stack of one trillion one-dollar bills! 3 x 105 km is also about 3/4 the distance to the moon.

Friday 5 July 2013 - What is the volume of rubber warn off of all the tires in the U.S. in one year? [m3]

• Number of tires based on the number of people

• Volume of rubber in the tread of a tire

• Time it takes for rubber to wear down

Result: answers varied between 105 - 106 m3

What does this mean? If we spread this volume of rubber over the entire United States, it would be about 1 micron high!

Tuesday 2 July 2013 - How tall is a stack of one trillion one-dollar bills? [km]

• Thickness of one one-dollar bill: 2 x 10-4 m

This is based on the thickness and total number of pages in the fifth Harry Potter book.

Result: 2 x 105 km

What does this mean? This stack of bills could be wrapped around the circumference of the earth about 5 times!

Useful Reference

Weinstein, L., Guesstimation 2.0: solving today’s problems on the back of a napkin. Princeton: Princeton University Press. 2012.