“I’m All About That Scale, ‘bout That Scale . . .”

Modeling Exponential Growth

Teaching Contemporary Mathematics 2015Kevin Bartkovich

Phillips Exeter Academykbartkovich@exeter.edu

How can you group these 9 things into 3 sets of 3?

Cha

rles R

ichte

r

Moore’s Law

New

ton

Pressure

Carbon

Savings Account

Population

How can you group these 9 things into 3 sets of 3?

Cha

rles R

ichte

r

Moore’s Law

New

ton

Pressure

Carbon

Savings Account

Population

Log Scales

ExponentialGrowth

Exponential Decay

Log Scales - Earthquakes

[From Exeter Math Book 3, page 71.]

Richter Scale

An increase by 1 in the rating R means that the energy released increases by a factor of 103/2, or about 31.6 times as much energy.

A difference of 2 in magnitude is equivalent to a factor of about 1000 in the energy released.

http://newsimg.bbc.co.uk/media/images/47199000/gif/_47199970_richter_earthquake4_466.gif

Frequency of Occurrence of EarthquakesMagnitude Average Annually

8 and higher 1 ¹7 - 7.9 15 ¹6 - 6.9 134 ²5 - 5.9 1319 ²4 - 4.9 13,000 (estimated)

3 - 3.9 130,000 (estimated)

2 - 2.9 1,300,000 (estimated)

¹ Based on observations since 1900. These numbers have been recently updated, based on data from the Centennial catalog (from 1900 to 1999) and the PDE (since 2000). ² Based on observations since 1990

Frequency vs. Magnitude of Earthquakes

[ http://earthquake.usgs.gov/earthquakes/eqarchives/year/eqstats.php ]

Log Scales - Decibels

Exeter Math Book 3, page 74

[ http://esm.versar.com/pprp/ceir15/Report_3_1_4.htm ]

Decibels:

TABLE G-16 - PERMISSIBLE NOISE EXPOSURES_________________________________________ | Duration per day, hours | Sound level______________________|__________________ |8 . . . . . . . . . . . . . . . . . . . . . .| 906 . . . . . . . . . . . . . . . . . . . . . .| 924 . . . . . . . . . . . . . . . . . . . . . .| 953 . . . . . . . . . . . . . . . . . . . . . .| 972 . . . . . . . . . . . . . . . . . . . . . .| 1001 1/2. . . . . . . . . . . . . . . . . . .| 1021 . . . . . . . . . . . . . . . . . . . . . .| 1051/2 . . . . . . . . . . . . . . . . . . . .| 1101/4 or less. . . . . . . . . . . . . .| 115______________________|__________________

OSHA Standards

https://www.osha.gov/pls/oshaweb/owadisp.show_document?p_table=standards&p_id=9735

OSHA Model

One more familiar log scale:

http://en.wikipedia.org/wiki/PH#mediaviewer/File:216_pH_Scale-01.jpg

Historic Value of Dow Jones Industrial Average

1880 1900 1920 1940 1960 1980 2000 20200

2000

4000

6000

8000

10000

12000

14000

16000

Value of DJIA with log scale

1880 1900 1920 1940 1960 1980 2000 20201

10

100

1000

10000

100000

One day sitting in assembly . . .

Figure 5. The observed probability (local p-value) that the background-only hypothesis would yield the same or more events as are seen in the CMS data, as a function of the SM Higgs boson mass for the five channels considered. The solid black line shows the combined local p-value for all channels.

Dice Simulation and Exponential Data

• Start with at least 50 dice• Pick a number from 1 to 6• After each toss of the dice, remove all the dice that show the

chosen number• Plot the number of dice left versus the number of tosses• Re-express (and linearize) the data by plotting the log of the

number of dice left vs. number of tosses• Find a model for the original data set by “un-doing” the semi-

log transformation

Population Growth in East Africa

• The population data for countries in East Africa show an exponential trend

• Notice how changing the vertical scale to “log” produces a semi-log graph, which shows a linear trend in each country

• The slope of the linearized data gives the constant percentage growth rate

http://www.google.com/publicdata/explore?ds=d5bncppjof8f9_&ctype=l&strail=false&bcs=d&nselm=h&met_y=sp_pop_totl&scale_y=lin&ind_y=false&rdim=region&idim=country:UGA:KEN:TZA:RWA:BDI:ZAR&ifdim=region&hl=en&dl=en&ind=false&icfg

World Population with Log Scale

World Population

• Recent data is not exponential• Historically there were times when world population

was growing exponentially• Other models need to be looked at for projections

based on recent trends

http://en.wikipedia.org/wiki/World_population#mediaviewer/File:World_population_growth_(lin-log_scale).png

Why is the slope of a line in a semi-log transformation equal to the growth rate of the exponential?

This last equation is linear in (x, log y).

Modeling Atmospheric CO2 Concentration

BAC and THC: Two Very Different Models

• Blood alcohol content (BAC) is proportional to the number of drinks and inversely proportional to the weight of the person.

• BAC declines at a rate of .015 per hour in men and at a rate of .017 in women.

• How does BAC vary over time?

• Make two graphs, one for a 160-lb man and one for a 120-lb woman, showing BAC vs. time (hours) for 1, 2, 3, 4, and 5 drinks.

Reference: http://www.clemson.edu/campus-life/campus-services/redfern/alcohol/bac.html

Table of BAC Relative to Body Weight and Number of Drinks

Plot of BAC vs. time in hours

THC concentration

• The amount of THC in the blood spikes quickly (within minutes), then decays exponentially.

• A typical concentration of THC after smoking marijuana is 160 ng/ml (nanograms per milliliter) within minutes, then reducing to 5 ng/ml in 3 hours. What exponential function models THC concentration? What is the half-life of THC in the blood?

• Certain states are considering laws that would set 5 ng/ml THC as the legal maximum for driving while impaired. How long would it take to reach 5 ng/ml if two doses of marijuana were taken 1 hour apart, each one raising the THC level by 100 ng/ml?

TCH Concentration vs. Time

Kepler and Logarithms

Exeter Math Book 3, page 56

The “Power” of Compound Interest And Saving for Retirement

0 5 10 15 20 250

1020304050607080

Model with Original Data For Dice Experiment

Number RemainingModel original data

1880 1900 1920 1940 1960 1980 2000 20200

2000400060008000

10000120001400016000

Frequency of Occu

rrence of E

arthquakes

Magnitude

A

verage Annually

8 and higher

1 ¹

7 - 7.9

15 ¹

6 - 6.9

134 ²

5 - 5.9

1319 ²

4 - 4.9

1

3,000

(estimated)

3 - 3.9

1

30,000

(estimated)

2 - 2.9

1,300,000

(estimated)

TABLE G-16 - PERMISSIBLE NOISE EXPOSURES

_________________________________________

|

Duration per day, hours | Sound level

______________________|__________________

|

8 . . . . . . . . . . . . . . . . . . . . . .| 90

6 . . . . . . . . . . . . . . . . . . . . . .| 92

4 . . . . . . . . . . . . . . . . . . . . . .| 95

3 . . . . . . . . . . . . . . . . . . . . . .| 97

2 . . . . . . . . . . . . . . . . . . . . . .| 100

1 1/2. . . . . . . . . . . . . . . . . . .| 102

1 . . . . . . . . . . . . . . . . . . . . . .| 105

1/2 . . . . . . . . . . . . . . . . . . . .| 110

1/4 or less. . . . . . . . . . . . . .| 115

______________________|__________________

BAC and THC1880 1900 1920 1940 1960 1980 2000 20201

10

100

1000

10000

100000

LOG SCALES

EXPONENTIAL GROWTH

EXPONENTIAL DECAY

World Population with Log Scale

RETIREMENT SAVINGS

KEPLER

DJIA

The greatest shortcoming of the human race is our inability to understand the exponential function.

Dr. Albert A. Bartlett, Emeritus Professor of Physics, University of Colorado; World Population Balance Board of Advisors

[ http://www.worldpopulationbalance.org/quotes ]