Back of the Envelope (BOTE) Calculations.pdf

7/28/2019 Back of the Envelope (BOTE) Calculations.pdf

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Engin eering 1000

Add it ion: Back of the Envelope (BOTE) Calculat ions

Adapted from original course notes developed by R. Hornsey

B. M. Quine

B. M. QuineBOTE 2

Outl ine

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

useful?

n Examples

n General approaches

n Excercises


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B. M. QuineBOTE 3

What & Why?

n Often during brainstorming, discussions, out in the field

engineers need to make rapid estimates

n to eliminate candidate solutions

n establish feasibility

n sketch out potential paths to a solution

n Although most engineers remember key numbers related to

their field, no-one has every detail at their fingertips

n Hence we need to estimate not only the values of numbers we

need, but which numbers are appropriate, and how to perform

the calculation

n the emphasis here is on order of magnitude estimates to the

nearest factor of 10

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

B. M. QuineBOTE 4

Accu racy of calcu lat ions

n A well-known curve of a calculations 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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B. M. QuineBOTE 7

How m any piano tuners in NYC?

n Approximately how many people are in New York City?

n 10,000,000

n Does every individual own a piano?

n No

n Would it be reasonable to assert that "individuals don't tend to

own pianos; families do?

n Yes.

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

people?

n Perhaps there are 2,000,000 families in NYC.

n Does every family own a piano?

n No.

n Perhaps one out of every five does.

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

B. M. QuineBOTE 8

n How many piano tuners are needed for 400,000 pianos?

n 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.

n How many piano tunings can one piano tuner do?

n 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.

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

tuners.


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B. M. QuineBOTE 9

General p r inciplesn When you use back-of-the-envelope calculations, be sure to

recall Einstein's famous advice.

n everything should be made as simple as possible, but no simpler

n Dont worry about specific factors

n round to the nearest sensible value

n corollary: dont make numbers more precise than is necessary

n Guess numbers you dont know

n but try to make your guesses good ones and within the bounds of

common sense

n common sense requires some education the accuracy of common

sense increases with experience

n Adjust geometry etc. to suit you

n

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

n e.g. use ratios

n assume unknown value is same as a similar known quantity

B. M. QuineBOTE 10

General pr inc ip les ctd

n Use the principle of conservation

n what goes in must either come out or stay inside

n things are not generally destroyed, so work out where they have gone

n Ensure formulas are dimensionally correct

n i.e. an expression to tell you the length of something must have overalldimensions of metres [THIS IS VERY POWERFUL TECHNIQUE]

n Apply a plausibility filter

n if an answer seems unbelievable, it probably is

n you can usually set a range of possible/reasonable values for aquantity that will indicate a major mistake (e.g. speed cannot be fasterthan speed of light!)

n Bound or threshold problems to scope design solutions

n Eg: Will it fit in? No it is 100 times too large even with optimistic values.

n Consider your results in context with the assumptions youhave made.

n Use two different methods to contrast Rough Order ofMagnitude ROM estimates (eg estimation and comparison).


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B. M. QuineBOTE 11

G. I. Taylor s famous 1947 Analysis us ing radius ,

energy, densi ty, t ime

Published U.S.

Atomic Bomb

was 18 kilotondevice a fact

considered top

secret at the time!

B. M. QuineBOTE 12

Excercises

n What area of space-based solar panels would it take to replace

Ontarios existing hydro (coal, gas, nuclear) power generating

capacity?

n How many air molecules do you breathe in your life that were

breathed by Pythagoras?


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B. M. QuineBOTE 13

Uncertainty

n Once you have a method to solve the problem, you can include

best- or worst-case estimates

n e.g. how many light bulbs are there in the U.S.?

n somewhere between 108 and 109 people

n not less than 1 light bulbs/person

n likely not more than 103 /person

n so the range of the answer is from 108 to 1012 light bulbs

n The bounding box

n between what values are we sure the answer lies

n to get the largest possible overestimate, multiply all the largest possible

values and divide by all the smallest possible values

n and vice versa for the lowest possible underestimate

n The likely box

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

values

B. M. QuineBOTE 14

Landmarksn Why create landmarks? To have stepping-stones for thought.

n For thought and imagination to move easily, one needs

handholds, markers

n bits of easily accessible knowledge scattered across the landscape.

Making its shape visible, and providing places to stand.

n Why remember landmarks when we have them in books?

n to carry them with us.

n Books can be a source of landmarks, and a place to keep

landmarks you don't often use

n also a place to keep more detailed versions of the landmarks you carryaround

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

where you left it, forgotten at home

n either it is ready at hand, or you need to use something else.

n 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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B. M. QuineBOTE 15

Example

n 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

B. M. QuineBOTE 16

Solut ionn 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 familiesbuy 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 .

n 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.n 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.

n 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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Back of the Envelope (BOTE) Calculations.pdf