Transcript Section 3B Putting Numbers in Perspective

Section 3B Part II
Putting Numbers in
Perspective
Pages 152-164
3-B
To appreciate really, really large numbers
and really, really small numbers, we must
“put them in perspective.”
Three Techniques
• Perspective through Estimation
• Perspective through Comparisons
• Perspective through Scaling
3-B
1. Perspective Through Estimation
pg155 How high is 1000 feet?
Each story ≈ 10 feet (floor to ceiling)
So 100-story building ≈ 1000 feet
(Empire State Building is 102 stories and 1250
feet so…. one story = 10-12 feet)
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31/165 Which is larger – the height of a 10-story apartment
building, or the length of a football field? By how much?
Each story ≈ 10-12 feet high
10 story building ≈ 10 stories  10-12 ft/story = 100 - 120 ft
Football field = 100 yards = 100 yd × 3ft/yd
= 300 ft
The football field is almost 3 times as long as the 10-story
building is high.
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Definition: An order of magnitude estimate
specifies only a broad range of values, such as
“in the tens of thousands” or “in the millions.”
The population of the U.S. is “on the order of
300 million”.
(2000 census says 281 million)
Ex3/155 Make an order of magnitude estimate of
the total amount of money spent each year on
ice cream in the US.
Assumptions:
≈ $1 per serving
≈ 1 serving of ice cream per week.
52 weeks per year
≈ $52 per person per year
≈ 300 million Americans
So, 300 million  52 = 300 x 106 x 5.2 x 10
= 3 x 108 x 5.2 x 10
= 15.6 x 109
≈ $15.6 billion spent by Americans for ice cream each
year.
3-B
2. Perspective Through Comparison
(compare to something easily understandable)
pg 156 How much is $100 billion?
$100 billion (= $100  109 )
= $1.0×1011
How long would it take to count this many $1 bills –
at a rate of 1 bill per second?
 1 min   1 hr   1 day   1 yr 
1.0  10 sec  




60
sec
60min
24hr
365
days

 
 
 

11
 3,171 years
3-B
Comparison for Unfamiliar Units
41/166 How many average candy
bars would you have to eat to
supply the energy needed for four
hours of running?
One hour of running requires 4 x 106 joules.
Four hours of running requires 16 x 106 joules.
Metabolism of one candy bar supplies 1 x 106 joules.
Metabolism of sixteen candy bars supplies 16 x 106 joules.
3-B
Ex4/157 Compare the U.S. population to
world population and the U.S. energy
consumption to world energy
consumption.
Population
U.S.
300 million
Energy
Consumption
1×1020 joules
World
6 billion
5×1020 joules
(300 million)/(6 billion) = (3x108)/(6x109) = .5x10(-1) = .05 = 5%
The US population is approximately 5% of the 20
world population.
U.S.energy 110
1


10
World
energy 5 10
5
(1x1020)/(5x1020) = 0.20
= 20%
The US population consumes 20% of the world’s energy.
3. Perspective Through
Scaling
3 common ways of expressing scales
• Verbally (1 in = 1 mi)
• Graphically (marked mini-ruler)
• Scale ratio (1 to 92,160)
3-B
3-B
Graphically (marked mini-ruler)
3-B
Finding the scale ratio:
EX6/159 A city map states 1 in = 1 mi.
What is the scale ratio for the map?
1 mile x 5280 ft/mile x 12 in/ft = 63,360 inches
One map inch represents 63360 ground inches.
The scale ratio for this map is 1 to 63360.
Actual distances are 63360 times map distances.
Timelines
EX9/161 Human civilization, at least since the time of ancient
Egypt is on the order of 5000 years old. The age of the
Earth is on the order of 5 billion years. Suppose we use the
length of a football field or about 100 meters as a timeline
to represent the age of the Earth. How far from the line’s
end would human civilization begin.
100 meters = 5 x 109 years and 1 meter = 5 x 107 years
5000 years = 5 x 103 years x 1 meter
5x107 years
= 1 x 10(-4) meters
= .0001 meters = 0.1 mm
WOW! On a timeline where the Earth’s history stretches
the length of a football field, human civilization appears
only in the final tenth of a millimeter
3-B
More Practice
pg 161 How large is a University
with 25,000 students?
IDEA: Suppose the President plans to meet with 5
students for lunch, 5 times per week. How long will it
take her to meet with the entire student body?
She meets with 5x5 = 25 students/week
25,000/25 = 1000 weeks to meet entire student body
1000 weeks/(50 weeks per year)
= 20 years of lunches !!
More Practice
pg 161 How small is an atom?
IDEA: How many can we place on a meter stick?
A typical atom has a diameter of about 10-10 meters
We could fit 1010 = 10 x 109 = 10 billion atoms along a meter stick.
1010 atoms = 1 meter = 100 centimeters
108 atoms = 1 centimeter
so 100 million atoms fill a 1 cm line.
More Practice
39/165 Make an order of magnitude
estimate for the amount of gas an
average adult driver uses per year.
61/166 There are approximately 2.2 million marriages
in the US each year. Express this quantity in
marriages per day.
65/166 The national debt was $6.9 trillion at the end
of 2003. Express this quantity as the height of a stack
of one-dollar bills. (10 bills = 1 mm).
3-B
Homework
Pages 165-167
# 32,35,36,42,56,59,62,63,64
Important Numbers
• Current US Population ≈ 281 million (300 million)
• Approximate distance across U.S = 4000 km
• We breathe about 10 times per minute.
• The average person can walk 10-15 miles per day.