Section 3B Putting Numbers in Perspective Reprise Pages

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 3-B

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. 3-B

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 ≈ feet high Total height ≈ 10  10 ft = 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. 3-B

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

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 × 10 6) × (5.2 × 10) = (3 x 10 8 ) × (5.2 × 10) = 15.6 × 10 9 ≈ $15.6 billion spent by Americans for ice cream each year.

2. Perspective Through Comparison $100 billion (= $100  10 9 ) = $1.0×10 11 How long would it take to count this many $1 bills – at a rate of 1 bill per second? 3-B

Selected Energy Comparisons 3-B

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 10 6 joules. Metabolism of one candy bar supplies 1 x 10 6 joules. Metabolism of sixteen candy bars supplies 16 x 10 6 joules. Four hours of running requires 16 x 10 6 joules.

U.S. versus World Energy Consumption PopulationEnergy Consumption U.S.300 million 1×10 20 joules World6 billion5×10 20 joules 3-B

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) 3-B

Perspective Through Scaling 3-B

Finding the scale ratio: On the map, 1 in. represents 16/11 mi Convert 16/11 mi to inches: Ratio is 1 to 92, no units on scale ratios! 3-B

Finding the scale ratio: A city map states 1 in = 1 mi. What is the scale ratio for the map? 3-B One map inch represents ground inches. The scale ratio for this map is 1 to Actual distances are times map distances.

Case Studies How 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? 3-B

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) = years of coffees! 3-B

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.) 3-B

How large is a billion dollars? $1 billion / $50,000 per worker = 20,000 employees 3-B

How large is a billion dollars? Another viewpoint: Sports star - salary = $1 million / year ( = $1 x 10 6 / yr) How long (in years) will it take you to become a billionaire? 3-B

How large is a billion dollars? Another viewpoint: # years = $1 billion / $1 million = 1   10 6 = 10 3 years = 1,000 years 3-B

How small is an atom? A typical atom has a diameter of about 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 (1/10 2 ) of a meter, so we could fit /10 2 =10 8 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)! 3-B

Homework for Friday Pages # 32,36,56,62,63,64,67 3-B 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 miles per day. A stack of ten $1 bills is approximately 1 mm thick.