Section 3BPutting Numbers in

PerspectiveReprise

Pages 152-164

Giving Meaning to Numbers

To appreciate really, really large numbers and really, really small numbers, we must “put them in perspective.”

• Perspective through Estimation• Perspective through

Comparisons• Perspective through Scaling

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1. Perspective Through Estimation

How high is 1000 feet?

Each story ≈ 10 feet (floor to ceiling)

So 1000 ft ≈ 100-story building

Empire State Building is 102 stories, 1250 feet.

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Perspective Through Estimation

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 highTotal height ≈ 10 10 ft = 100 - 120 ft high

Football field = 100 yards = 100 yd × 3ft/yd = 300 ft long

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)

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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 × 106) × (5.2 × 10) = (3 x 108) × (5.2 × 10) = 15.6 × 109

≈ $15.6 billion spent by Americans for ice cream each year.

2. Perspective Through Comparison$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

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Selected Energy Comparisons

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On average, how many 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.

Metabolism of one candy bar supplies 1 x 106 joules.Metabolism of sixteen candy bars supplies 16 x 106 joules.

Four hours of running requires 16 x 106 joules.

U.S. versus World Energy Consumption

Population

Energy Consumption

U.S. 300 million

1×1020 joules

World 6 billion 5×1020 joules6

9

U.S.pop. 300 10.05 5%

World pop. 6 10

20

20

U.S.energy 1 10 1.2 20%

World energy 5 10 5

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3. Perspective Through Scaling

3 common ways of expressing scales

• Verbally (1 in = 16/11 mi)• Graphically (marked mini-ruler) • Scale ratio (1 to 92,160)

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Perspective Through Scaling

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Finding the scale ratio:

On the map, 1 in. represents 16/11 mi

Convert 16/11 mi to inches:

Ratio is 1 to 92,160--- no units on scale ratios!

16 16 5280ft 12 inmi mi

11 11 1mi 1ft

92,160 in.

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Finding the scale ratio:

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

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One map inch represents 63360 ground inches.

The scale ratio for this map is 1 to 63360.

Actual distances are 63360 times map distances.

5280ft 12 in1mi 1mi

1mi 1ft

63,360 in

Case StudiesHow large is a University with 26,000

students?

President plans to meet with 5 students for coffee, 5 times per week.

How long will it take her to meet with the entire student body?

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26,000 students

She meets with 5x5 = 25 students/week

26,000/25 = 1040 weeks for the entire student body

1040 weeks/(48 weeks per year)= 21.67 years of coffees!

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Case Studies

• How large is a billion dollars?• How many people could you employ

per year with this much money?

• Assume each worker costs $50,000/year ($25,000 salary + health insurance, etc.)

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How large is a billion dollars?

$1 billion / $50,000 per worker

= 20,000 employees

9

4$102 10 employees

$50,000

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How large is a billion dollars?Another viewpoint:

Sports star - salary = $1 million / year

( = $1 x 106 / yr)

How long (in years) will it take you to become a billionaire?

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How large is a billion dollars?Another viewpoint:

# years = $1 billion / $1 million = 1 109

1 106

= 103 years = 1,000 years

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How small is an atom?A typical atom has a diameter of about 10-10

meters (one ten billionth of a meter)

- so we could fit 10 billion of them in a line along a meter stick.

A centimeter is 1/100 (1/102) of a meter, so we could fit 1010/102 =108 or 100 million atoms in a 1cm line.

Thus the ≈300 million US people could fit in a 3cm line (if we could shrink them to the size of atoms)!

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Homework for Friday

Pages 165-167# 32,36,56,62,63,64,67

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Important Numbers• Current US Population ≈ 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.• A stack of ten $1 bills is approximately 1 mm thick.