Engineering 1000Addition: Back of the Envelope (BOTE) Calculations

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Outline What is a back-of-the-envelope calculation and why is it

useful? Examples General approaches Excercises

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What & Why? Often – during brainstorming, discussions, out in the field –

engineers need to make rapid estimates to eliminate candidate solutions establish feasibility sketch out potential paths to a solution

Although most engineers remember key numbers related to their field, no-one has every detail at their fingertips

Hence we need to estimate not only the values of numbers we need, but which numbers are appropriate, and how to perform the calculation

the emphasis here is on “order of magnitude” estimates – to the nearest factor of 10

it is also important to remember that these are rough estimates and to place only appropriate reliance on the results

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Accuracy of calculations A well-known curve of a calculation’s accuracy versus mental

effort goes like:

% error

effort

errors in various assumptions cancel out so rapid apparent improvement is made

better understanding may actually make things worse

hard work means that the model yields improved results

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Fermi Questions About 50 years ago, Enrico Fermi asked his physics students

at the University of Chicago "How many piano tuners are there in Chicago?" his idea was to encourage the students to think about the process of

estimating the answer without any specialised knowledge a Fermi question requires estimation of physical quantities to arrive at

an answer. Throughout his work, Fermi was legendary for being able to

figure out things in his head, using information that initially seems too meager for a quantitative result

he used a process of "zeroing in" on problems by saying that the value in question was certainly larger than one number and less than some other amount. He would proceed through a problem in that fashion and, in the end, have a quantified answer within identified limits.

In a Fermi question, the goal is to get an answer to an order of magnitude (typically a power of ten) by making reasonable assumptions about the situation, not necessarily relying upon definite knowledge for an "exact" answer.

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A Fermi question is posed with limited information given. how many water balloons would it take to fill the school gymnasium? how many piano tuners are there in New York City? what is the mass in kilograms of the student body in your school?

A Fermi question requires that students ask many more questions.

how big is a water balloon? what are the approximate dimensitons dimensions of the gym? what measurment must be estimated using the dimensions of the gym? ... and the list goes on.

A Fermi question demands communication. A Fermi question utilizes estimation. A Fermi question emphasizes process rather than "the"

answer.

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How many piano tuners in NYC? Approximately how many people are in New York City?

10,000,000 Does every individual own a piano?

No Would it be reasonable to assert that "individuals don't tend to

own pianos; families do? Yes.

About how many families are there in a city of 10 million people?

Perhaps there are 2,000,000 families in NYC. Does every family own a piano?

No. Perhaps one out of every five does.

That would mean there are about 400,000 pianos in NYC.

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How many piano tuners are needed for 400,000 pianos? Some people never get around to tuning their piano; some people tune

their piano every month. If we assume that "on the average" every piano gets tuned once a year, then there are 400,000 "piano tunings" every year.

How many piano tunings can one piano tuner do? Let's assume that the average piano tuner can tune four pianos a day.

Also assume that there are 200 working days per year. That means that every tuner can tune about 800 pianos per year.

How many piano tuners are needed in NYC? The number of tuners is approximately 400,000/800 or 500 piano

tuners.

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General principles When you use back-of-the-envelope calculations, be sure to

recall Einstein's famous advice. “everything should be made as simple as possible, but no simpler”

Don’t worry about factors of 2, π, etc. round to the nearest 0, 5, 10 corollary: don’t make numbers more precise than is necessary

Guess numbers you don’t know but try to make your guesses good ones and within the bounds of

common sense common sense requires some education – the accuracy of common

sense increases with experience Adjust geometry etc. to suit you

assume a human is spherical if it helps Extrapolate from what you do know

e.g. use ratios assume unknown value is same as a similar known quantity

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General principles ctd … Use the principle of conservation

what goes in must either come out or stay inside things are not generally destroyed, so work out where they have gone

Ensure formulas are dimensionally correct i.e. an expression to tell you the length of something must have overall

dimensions of metres Apply a ‘plausibility’ filter

if an answer seems unbelievable, it probably is you can usually set a range of possible/reasonable values for a quantity

that will indicate a major mistake (e.g. speed cannot be faster than speed of light!)

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Excercises What area of solar panels would it take to replace Ontario’s

existing hydro (coal, gas, nuclear) power generating capacity? How many ‘air’ molecules do you breathe in your life that were

breathed by Pythagoras?

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Uncertainty Once you have a method to solve the problem, you can

include best- or worst-case estimates e.g. how many light bulbs are there in the U.S.?

somewhere between 108 and 109 people not less than 1 light bulbs/person likely not more than 103 /person so the range of the answer is from 108 to 1012 light bulbs

The bounding box between what values are we sure the answer lies to get the largest possible overestimate, multiply all the largest possible

values and divide by all the smallest possible values and vice versa for the lowest possible underestimate

The likely box similar to the bounding box but using the largest and smallest likely

values

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Landmarks Why create landmarks? To have stepping-stones for thought. For thought and imagination to move easily, one needs

handholds, markers bits of easily accessible knowledge scattered across the landscape.

Making its shape visible, and providing places to stand. Why remember landmarks when we have them in books?

to carry them with us. Books can be a source of landmarks, and a place to keep

landmarks you don't often use also a place to keep more detailed versions of the landmarks you carry

around but when something would be useful, it doesn't help much to know just

where you left it, forgotten at home either it is ready at hand, or you need to use something else.

Over time we change what we carry with us. Choose knowledge which helps with our interests and current questions about the world

http://www.vendian.org/envelope/dir0/landmarks.html

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Example How much paper does The New York Times use in a week? A

paper company that wishes to make a bid to become their sole supplier needs to know whether they have enough current capacity. If the company were to store a two-week supply of newspaper, will their empty 14,000 square foot warehouse be big enough?

http://www.nap.edu/html/hs_math/be.html

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Solution The New York Times. Answering this question requires two preliminary estimates: the circulation of

The New York Times and the size of the newspaper. The answers will probably be different on Sundays. Though The New York Times is a national newspaper, the number of subscribers outside the New York metropolitan area is probably small compared to the number inside. The population of the New York metropolitan area is roughly ten million people. Since most families buy at most one copy, and not all families buy The New York Times, the circulation might be about 1 million newspapers each day. (A circulation of 500,000 seems too small and 2 million seems too big.) The Sunday and weekday editions probably have different circulations, but assume that they are the same since they probably differ by less than a factor of two--much less than an order of magnitude. When folded, a weekday edition of the paper measures about 1/2 inch thick, a little more than 1 foot long, and about 1 foot wide. A Sunday edition of the paper is the same width and length, but perhaps 2 inches thick. For a week, then, the papers would stack 6 x 1 / 2 + 2 = 5 inches thick, for a total volume of about 1 ft x 1 ft x 5 / 12 ft x 0.5 ft3 .

The whole circulation, then, would require about 1/2 million cubic feet of paper per week, or about 1 million cubic feet for a two-week supply.

Is the company's warehouse big enough? The paper will come on rolls, but to make the estimates easy, assume it is stacked. If it were stacked 10 feet high, the supply would require 100,000 square feet of floor space. The company's 14,000 square foot storage facility will probably not be big enough as its size differs by almost an order of magnitude from the estimate. The circulation estimate and the size of the newspaper estimate should each be within a factor of 2, implying that the 100,000 square foot estimate is off by at most a factor of 4--less than an order of magnitude.

How big a warehouse is needed? An acre is 43,560 square feet so about two acres of land is needed. Alternatively, a warehouse measuring 300 ft x 300 ft (the length of a football field in both directions) would contain 90,000 square feet of floor space, giving a rough idea of the size.

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Further practise questions

1. How many jelly beans fill a one-litre jar?

2. What is the mass in kilograms of the student body in your school?

3. How many golf balls will fill in a suitcase?

4. How many gallons of gasoline are used by cars each year in the United States?

5. How high would the stack reach if you piled on trillion dollar bills in a single stack?

6. Approximately what fraction of the area of the continental United States is covered by automobiles?

7. How many hairs are on your head?

8. What is the weight of solid garbage thrown away by American families every year?

9. If your life earnings were doled out to you at a certain rate per hour for every hour of your life, how much is your time worth?

10. How many cells are there in the human body?

11. How many individual frames of film are needed for a feature-length film? How long is such a film?

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