“I’m All About That Scale, ‘bout That Scale . . .”
Modeling Exponential Growth
Teaching Contemporary Mathematics 2015Kevin Bartkovich
Phillips Exeter [email protected]
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 ]