Estimation Problems

Chapter 5

ESTIMATION

Estimation Problems

 Number of hair on your head?

 Drops of water in a lake?

 HafiZ of Quran in the world?

• How many cubic yards of concrete are needed to pave one mile of interstate highway (two lanes each direction) ?

• How many feet of wire are needed to connect the lighting systems in an

automobile?

Estimation Problems

Enrico Fermi

 Brilliant scientist and engineer

 Worked in Manhattan Project

 Development of Nuclear weapons

 Witnessed Trinity Test

 First atomic bomb explosion

• Nobel laureate

• Taught at University of Chicago

• Gave students problems:

• Much information missing

• Solution seemed impossible

• Such problems: Fermi problems

Enrico Fermi

Trinity Test

• After the explosion

•Brighter than in full daylight

• After a few seconds

• Rising flames lost their brightness

• Huge pillar of smoke

• Rose rapidly beyond the clouds

• A height of the order of 30,000 feet

Trinity Test

•About 40 seconds after the explosion

• Dropped small pieces of paper

• From a height of 6 feet

• Measured displacement of pieces of paper

• Shift was about 2½ meters

•Fermi estimated the intensity of explosion:

• As from ten thousand tons of T.N.T.

Trinity Test

• calculated the explosion to be 19 kilotons.

• By observing the behavior of falling bits of paper ten miles from the ground zero, Fermi's estimation of 10 kilotons was in error by less than a factor of 2.

• After the war, Fermi taught at the university of Chicago where he became famous for his unsolvable problems

FERMI PROBLEMS

• Fermi's problems require the person considering them to determine the answer with far less

information than would be necessary to calculate an accurate value.

Estimation

•Engineers are often faced with solving problems for which they do not have all the information.

They must be adept at making initial estimates.

• This skill helps them to identify critical information that is missing, develop their reasoning skill to solve problems.

•Out of fuel on the road!

•Gallons of gasoline carried to such vehicles each year in united states?

TYPICAL FERMI PROBLEM

Example 5-1

•US Population: 500,000,000

(Actual 317,254,000)

•Drivers? 7 per 10 persons?

•How many run out of gas per year?

“1 per 4 years per driver”

•24 of 25 bring gas to the car

•Amount of gas carried: 1.5 gallons

Example 5-1

• Drivers:

(5 x 10

⁸

people)(

^{7 drivers}_{10 people}

)

= 3.5 x 10

⁸

drivers

• Number out of gas per year:

(3.5 x 10

⁸

drivers)(

1 𝑜𝑢𝑡 𝑜𝑓 𝑔𝑎𝑠

(4 years)(1 driver)

)

= 8.75 x 10

⁷

Example 5-1

• Number bring gas to car per year (

8.75 𝑋 107 𝑜𝑢𝑡 𝑜𝑓 𝑔𝑎𝑠

year

)(

24 𝑏𝑟𝑖𝑛𝑔 𝑡𝑜 𝑐𝑎𝑟 25 out ofgas

)

= 8.4 x 10

⁷

• Amount of gas to cars (

^{8.4 𝑋 107}_year

)( 1.5 gallons )

= 1.26 x 10

^{8 𝑔𝑎𝑙𝑙𝑜𝑛𝑠}

year

5.1 GENERAL HINTS FOR ESTIMATION

• Try to determine the accuracy required. Is order of magnitude enough? He. about ± 25%?

• What level of accuracy is needed to calculate a satellite trajectory?

• Remember that a "ballpark" value for an input parameter is often go enough.

• What is the typical velocity of a car on the highway?

5.1 GENERAL HINTS FOR ESTIMATION

• Always ask yourself if it is better to err on the high side or the low side.

• Safety and practical considerations. Will a higher or lower estimate result in a safer or more reliable result?

• Don't get bogged down with second-order or minor effects.

• If estimating the mass of air in the classroom, do you need to correct for the presence of furniture?

Significant figures

Significant figures or “sig fig” are the digits considered reliable as a result of

measurement or calculation.

This is not to be confused with the number of digits or decimal places.

The number of decimal places is simply the number of digits to the right of the decimal point. Example 5.2 below illustrates these two concepts.

Significant figures

Decimal places are the number of digits to the right of the decimal point.

Significant figures are the digits considered reliable.

Significant in x & /

Multiplication and Division :

Final result should contain the same number of significant figures as the number with the fewest significant figures .

Example of Sig. Fig.

Example 5-3 :

(2.43)(17.675) = 42.95025 ≈ 43.0

* (2.43) has three significant figures.

* (17.675) has five significant figures.

* The answer must have three significant figures.

Example of Sig. Fig.

Example 5-4:

(2.479 hours)(60 minutes/hour)

= 148.74 ≈ 148.7

*(2.479 hours) has four significant figures.

*(60 minutes) has an exact conversion.

* The answer has four significant figures.

Addition and Subtraction

• The least precise number in the calculation.

• The least precise number:

With the lowest number of decimal places.

The Meaning of "Significant"

Example 5-5:

1725.463 + 489.2 + 16.73 = 1931.393

* 489.2 is the least precise.

• The answer should contain one decimal place

^:

1931.4

The Meaning of "Significant"

General hints for estimation

Try to determine the accuracy required. Is order of magnitude enough? How about (+ or -) 25%?

What level of accuracy is needed to calculate a satellite trajectory?

What level of accuracy is needed to determine the amount of paint needed to paint a specified classroom?

General hints for estimation

Remember that a “ ballpark “ value for an input parameter is often good enough

What is the square footage of a typical house?

What is the maximum high temperature to expect in Dallas, Texas in July?

What is the typical velocity of a car on the highway?

General hints for estimation

Always ask yourself if it is better to err:

on the high side?

on the low side ?

Safety and practical considerations

Weight a bridge can support?

it is better to err on the low side

the actual load it can carry > the estimate

REASONABLENESS

We consider two types of reasonableness in answer to problem in this section.

Physically reasonable.

Reasonable precision.

REASONABLENESS

When

is Something Physically Reasonable?

Here are a few hints to help you determine if a solution to problem is physically

reasonable.

REASONABLENESS

First ask yourself if the answer makes sense in the physical world.

You determine that wingspan of a new airplane to carry 200 passengers should be

four feet this is obvious rubbish.

REASONABLENESS

If the final answer is in units for which

you do not have an intuitive fell, convert to units for which you do have an

intuitive fell.

REASONABLENESS

you

are interested in what angle a smooth steel ramp must have before a wooden block will begin to slide down it. Your calculations show that the value is 0.55 radians.

REASONABLENESS

Is this reasonable? If you have a

better " feel " for degrees ، you should convert the value in radians to

degrees ، which gives 32 degrees ،

this value seems reasonable

5.3 REASONABLENESS

When Is an Answer Reasonably Precise?

First, we need to differentiate between the two terms accurate and precise^.

Accuracy : is a measure of how close a calculation or measurement is to the actual value.

Repeatability : is a measure of how close together multiple

measurements of the same parameter are, whether or not they are close to the actual value.

Precision : is a combination of accuracy and repeatability,.

5.3 REASONABLENESS

The figure shows all four

combinations of accuracy and repeatable.

•Neither repeatable nor accurate.

•Repeatable, but not accurate.

•Accurate, but not repeatable.

•Both repeatable and accurate.

This is called precise.

5.4 Notation

•In united states a period is used as decimal separator and comma is used as a digit group separator,

indicating groups of a thousands ( such as 5,245.25 ).

•In some countries, however this notation is reserved (5.245,25).

•And in other countries a space is used as the digit group

•separator ( 5 245.25).

• It is important to always consider the country of origin when interpreting written values.

5.4 Notation

• Engineering Notation versus Scientific Notation :

Engineering Notation :

###.###× 10^M

M is an integer multiple of 3

Scientific Notation :

#.###× 10^N N is an integer

EXAMPLE 5-9

Engineering Scientific

Standard

43.48 X 10⁶ 4.348 X 10⁷

43,480,000

306.0 X 10⁻⁹ 3.060 X 10⁻⁷

0.0000003060

9.86 X 10⁹ 9.86 X 10⁹

9,860,000,000

35.1 X 10⁻³ 3.51 X 10⁻²

0.0351

52.2 X 10⁻¹⁵ 5.22 X 10⁻¹⁴

0.0000000522

456.2 X 10⁶ 4.562 X 10⁸

456200

Comprehension Check 5-4 :

Express each of the following values in scientific & engineering notation.

Engineering Scientific

Standard

58.09 X 10⁶ 5.809 X 10⁷

58,093,099

4.581 X 10⁻³ 4.581 X 10⁻³

0.00458097

42.677 X 10⁶ 4.268 X 10⁷

42,677,000.99

5.4 Notation

• Calculator E-notation :

•Most scientific calculators use the capital letter E as shorthand for both scientific and engineering

notation.

•In general it is best to not use the E notation, thus 3.707 E – 5 should be written as 3.707 × 10^-5

NOTATION E

 It is important to always consider the country of origin when interpreting written values .

 Scientific notation is typically expressed in the form

#.### X10 .

 Engineering notation is expressed in the form ###.###

X10M , where M is an integer multiple of 3 .

 Most scientific calculators use the capital letter E as shorthand for both scientific and engineering notation when representing number .

NOTATION E

 To state the meaning of the letter E in English , it is read ( times 10 raised to the –).

 If the magnitude is greater than 10,000 or less than 0,0001 , you probably should consider using

exponential notation .

 It is never actually incorrect to use either exponential or standard notation , it is merely a matter of

readability .