Section 3B Part II Putting Numbers in Perspective Pages 152-171

Section 3B Part II Putting Numbers in Perspective

Pages 152-171

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

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)

3-B

pg157 How high is 1000 feet?

Example Which is larger – the height of a 10-story apartment building, or the length of a football field? By how much?

3-B

Each story ≈ 10-12 feet high10 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.

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)

3-B

Ex3/158 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 AmericansSo, 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.

2. Perspective Through Comparison(compare to something easily understandable)

pg 159 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?

11 1 min 1 hr 1 day 1 yr1.0 10 sec

60sec 60min 24hr 365

3,171 years

days

3-B

Comparison for Unfamiliar Units

3-B

Population Energy Consumption

U.S. 300 million 1×1020 joules

World

6 billion 5×1020 joules

20

10

U.S.energy 1 10 1

World energy 5 10 5

3-B

The US population is approximately 5% of the world population.

Ex4/160 Compare the U.S. population to world population and the U.S. energy consumption to world energy consumption.

(300 million)/(6 billion) = (3x108)/(6x109) = .5x10(-1) = .05 = 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)

Finding the scale ratio:

EX6/161 A city map states 1 in = 1 mi. What is the scale ratio for the map?

3-B

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

EX9/163 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

5000 years = 5 x 103 years x 100 meters 5x109 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

More Practicepg 164 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?

3-B

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

49/165 Make an order of magnitude estimate for the amount of gas an average adult driver uses per year.

71/170 Approximately 1.4 million violent crimes are reported to police every year. How many violent crimes are reported every hour?

75/166 The national debt was $8.5 trillion at the end of 2006. Express this quantity as the height of a stack of one-dollar bills. (10 dollar bills = 1 mm).

Homework

Pages 169-171# 42,46,50,65,69,72,74,77

3-B

Important Numbers• Current US Population ≈ 281 million (300 million)• Approximate distance across U.S = 4000 km • The average person can walk 10-15 miles per day.

Pankratova Legs 52 inches

He Pingping Height 29.37 inches

He Pingping’s Height is 56.5% of Pnakratova’s Leg Length

Percent Of 29.37/52 = .565