The Basics of Fermi Estimates
How many hours do all Americans
combined spend on the Internet in one year?
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
This week's estimation
If you paved a pathway to the moon, how
long would it take (in years) to:
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)
Result: Answers ranged between 2 x 104 to 6 x 106 kWh
What does this mean?
Friday 12 July 2013 - What is the maximum
resolution of the human 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).
Result: Answers ranged from 7 x 10-3 radians to 8 x 10-5 radians
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]
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]
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!
Weinstein, L., Guesstimation 2.0: solving today’s problems on the back of a napkin. Princeton: Princeton University Press. 2012.