A Sustainable by Pete Kaslik Pierce College, Fort Steilacoom
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Math In A Sustainable Society
Instructor Edition 2.2
by Pete Kaslik
Pierce College, Fort Steilacoom
Cover Photo
The photograph on the front cover was created by Chris Jordan (www.chrisjordan.com).
The original photo measures 8 feet by 11 feet, in three vertical panels. “The photo depicts 2.4
million pieces of plastic, equal to the estimated number of pounds of plastic pollution that enter
the world’s oceans every hour. All of the plastic in this image was collected from the Pacific
Ocean.” (Jordan, 2009)
Below is a close up view that shows the details of Mt. Fuji.
Math In A Sustainable Society
Instructors Edition 2.2
by Pete Kaslik
2010
Creative Commons License Attribution: Noncommercial 3.0
You are free to share or remix this work.
The edition number is written in the form M.m where M represents a major revision and m
represents minor revisions, such as typographical errors or the addition or correction of
problems. Minor corrections may result in some changes in page numbers compared to earlier
editions, but these changes should be minimal.
To Jean
Acknowledgements
I am very grateful for the willingness of Chris Jordan to allow me to include one of his
photos on the cover of this book. Issues of mass consumption result in numbers that are so big
that people have difficulty comprehending them. By taking those large numbers and reducing
them to a photo that represents only a small slice of time, the viewer is better able to grasp the
magnitude of the numbers and consequently the magnitude of human actions.
I am very appreciative of the help, support and editing by my wife, Jean. This book has
not had the benefit of professional editors so her feedback has been very valuable.
I appreciate my brother Jim Kaslik, for allowing the use of the picture of the dome home
he designed.
I wish to acknowledge David Lippman, chairman of the Math department at Pierce
College, for reviewing the book and for his support for my using it. I also wish to thank the
classes of students on whom I experimented with earlier editions of this book. Their suggestions
lead to the removal of some chapters that were less interesting and to the clarification of text,
activities and homework that were confusing.
Table of Contents
Chapter 0.5 Quantitative Assessment of the World Activity ................................................... 1
Chapter 1 Financial Survival ...................................................................................................... 5
SAVING IN ADVANCE ................................................................................................................... 6
GROWTH RATES AND GROWTH FACTORS ................................................................................ 9
FUTURE VALUE WITH ARITHMETIC GROWTH .......................................................................... 9
FUTURE VALUE WITH GEOMETRIC GROWTH ......................................................................... 10
THE COMPOUND INTEREST FORMULA .................................................................................... 11
ANNUAL PERCENTAGE YIELD .................................................................................................. 12
CONTINUOUS COMPOUNDING .................................................................................................. 14
GRAPHING EXPONENTIAL GROWTH ....................................................................................... 15
RULE OF 72 ................................................................................................................................ 20
SINKING FUND ............................................................................................................................ 23
BIG PURCHASES ......................................................................................................................... 27
MONTHLY PAYMENT ............................................................................................................ 27
AMORTIZATION .................................................................................................................... 28
THE EFFECT OF PREPAYMENT .............................................................................................. 29
CREDIT TROUBLE ....................................................................................................................... 32
Chapter 1.5 Sustainability ......................................................................................................... 41
If-Then Project ........................................................................................................................ 44
System Dynamics Models ....................................................................................................... 48
Chapter 2 Population Growth .................................................................................................. 53
MODELING POPULATION GROWTH ......................................................................................... 56
Chapter 3 The Algebra of Sustainability ................................................................................. 63
Chapter 4 Statistics .................................................................................................................... 85
EXPERIMENTS AND STUDIES .................................................................................................... 88
SAMPLING ................................................................................................................................. 89
PROBABILITY ............................................................................................................................ 93
SIMPLE PROBABILITY .......................................................................................................... 93
P(A OR B).............................................................................................................................. 94
P(A AND B) ........................................................................................................................... 95
USING DATA TO ANSWER QUESTIONS ..................................................................................... 96
GRAPHING QUANTITATIVE DATA ............................................................................................ 96
STATISTICS FOR QUANTITATIVE DATA ................................................................................... 98
STANDARD DEVIATION ......................................................................................................... 99
THEORY .................................................................................................................................. 103
SAMPLING DISTRIBUTION OF SAMPLE MEANS...................................................................... 105
CENTRAL LIMIT THEOREM.................................................................................................... 106
CONFIDENCE INTERVALS ....................................................................................................... 106
GRAPHS AND STATISTICS FOR QUALITATIVE (CATEGORICAL) DATA.... Error! Bookmark not
defined.
THEORY .................................................................................................................................. 112
CONFIDENCE INTERVAL FOR PROPORTIONS ......................................................................... 113
Chapter 5 System Dynamics Modeling .................................................................................. 123
COMPUTER MODELING .......................................................................................................... 127
Appendix .................................................................................................................................... 135
1
Chapter 0.5 Quantitative Assessment of the World Activity
Many students, particularly those taking algebra, wonder when they will use math in their
life. All too often, the justification involves things like giving change or balancing check books.
These are small applications that require only arithmetic. In this book, you will get to see some
larger applications of mathematics that will help you understand both personal and global issues
and the decisions that can be made as a result of this understanding. The issues that will be
addressed are those that cannot be understood without the mathematics.
To begin the process, you will look at a multitude of issues facing humanity. These
issues are provided on the next two pages. Look at the issues presented and find one to three
issues that interest you. In class, you will be able to sign up for one issue.
Your responsibility for this activity is to find one or two graphs that will help the class
understand the issue. An ideal graph will show the status of the issue today as well as
historically. In this context, today means during the last 1 to 5 years. Historically means over
the last few decades. Projections are acceptable too. If a temporal graph is not available, then a
spatial graph, such as one that shows the current status in the US and other countries should be
used. It is critical that either a temporal or spatial comparison is made as numbers in isolation do
not hold much meaning.
Ultimately, the class will watch the presentation and evaluate the topic on a scale of 0 to
4, in which 0 represents a critical state with a negative trend and 4 represents an excellent state
with an improving trend. Consider a critical state as one that could negatively affect us during
our lifetime. An excellent state is one that humanity should be proud of achieving.
Not all topics have the same importance. Besides scoring each topic, you will also give it
a weight using numbers between 0 and 3. A score of 0 means you don’t consider the topic to
have any importance at all to the well-being of life on earth. A score of 3 means you think the
topic is extremely important to life. After viewing all graphs, you will find the weighted mean of
your scores.
The graph should be copied into a Word document and sent to me as an email
attachment. I will compile the graphs. Each graph must include the source (URL). Graphs are
due to me by ______________.
This QAW project will be evaluated using the following criteria and points.
1. Provides useful information so audience can make a reasonable judgment (10)
2. Includes current status (5)
3. Includes temporal or spatial comparison (5)
4. Source (URL) provided with graph (2)
5. Graph submitted on time (6)
6. Presentation (competent and given when scheduled) (5)
7. Watch presentations and judge graphs (2)
Total: 35 points
Math In A Sustainable Society 2.2 – Instructors Manual
2
Suggestions for information your graph should show
Topics
Human Health and Well Being
1. Human Population WA
2. Human Population US
3. Human Population World
4. Poverty US
5. Poverty World
6. Violent Crimes US
7. Death by AIDS, Cars, Cancer
8. War
9. Prisons
10. Life Expectancy
11. Health Care Cost
12. Gender Relations
13. High School Graduation Rate
Food
14. Marine Fisheries
15.
16.
17.
18.
19.
Farms
Farmers Markets
Water Quality
Water Quantity
Bees
Changes in total population over time. You can also
show changes in the ethnic composition.
Changes in total population over time. You can also
show changes in the ethnic composition.
Besides showing changes in total population, show
changes in first world, second world and third world
countries.
Show how the value that indicates poverty and the
number of people in poverty has changed over time.
Show poverty levels in first world, second world and
third world countries.
Compare per capita violent crime rates in the US to
other nations.
Compare worldwide deaths from AIDS, traffic fatalities
and Cancer over time.
Deaths per year as a result of war or comparison of
number of deaths in various wars dating back to at least
the Civil War.
Show changes in prison populations. Show change in
cost to government.
Show life expectancy changes over time in the US.
Compare with other countries too.
Compare US to other nations
Compare equality of men and women by showing a
comparison for salaries for the same job, students in
higher education, proportion of women in positions of
authority (management, government)
Compare how graduation rates have changed over time.
Show changes in size of fish stocks from around the
world.
Show changes in the number of farms.
Show changes in the number of farmers markets
Show changes in water quality
Show major US aquifers and changes in water levels
Show changes in the honey bee population over the last
10 years.
Math In A Sustainable Society 2.2 – Instructors Manual
3
Environment and Energy
20. US Oil Production and
Consumption
21. World Oil Production and
Consumption
22. Oil Costs
23. Natural Gas Production
24. Coal Production
25. Driving Distances
26. Air Pollution
27. Climate Change
28. Electrical Energy
Financial
29. National Debt
30. Housing Size and Occupants
31. Housing Costs
32. Wealth Gap
33. College Education
34. Per Capita Income
35. Inflation
Show US oil production – explain peak oil
Show US oil consumption
Show world oil production
Show world oil consumption
Show costs per barrel and costs at the pump
Show world production and changes over time.
Show world production and changes over time.
Show the per capita annual distance driven in the US.
Compare to past years or to other countries.
Show changes in carbon dioxide and at least one other
including methane, ozone, acid rain
Show global temperature changes.
Compare the amount of energy produced by various
sources such as fossil fuels, nuclear, hydro, wind etc.
and show how that has changed over time.
Show changes in National Debt from at least the early
part of the 1900s
Show changes over time in the size and number of
people living in a house. Also, compare to other
countries.
Show changes in the cost of housing in US and
Washington
Show the Gini Coefficient for the US and other
countries. Explain the Gini Coefficient.
Show how costs have changed over time.
Show how income has changed over time. Compare it
with inflation.
Show US inflation rates over time.
Math In A Sustainable Society 2.2 – Instructors Manual
4
QAW Score Card
Name:
Human Health and Well Being
1. Human Population WA
2. Human Population US
3. Human Population World
4. Poverty US
5. Poverty World
6. Violent Crimes US
7. Death by AIDS, Cars, Cancer
8. War
9. Prisons
10. Life Expectancy
11. Health Care Costs
12. Gender Relations
13. High School Graduation Rate
Food
14. Marine Fisheries
15. Farms
16. Farmers Markets
17. Water Quality
18. Water Quantity
19. Bees
Environment and Energy
20. US Oil
21. World Oil
22. Oil Costs
23. Natural Gas Production
24. Coal Production
25. Driving Distances
26. Air Pollution
27. Climate Change
28. Electrical Energy
Financial
29. National Debt
30. Housing Size and Occupants
31. Housing Costs
32. Wealth Gap
33. College Education
34. Per Capita Income
35. Inflation
Total
Weighted Mean
Importance
Weight (0-3)
Score (0-4)
Multiply the Weight
times the Score
∑W =
xxxxxxxxx
xxxxxxxxxxxxxxx
∑(W*S)/∑W =
∑(W*S)=
xxxxxxxxxxxxxx
Math In A Sustainable Society 2.2 – Instructors Manual
5
Chapter 1 Financial Survival
We will start understanding the importance of math in society by looking at an issue of
concern to most people. That issue is money. Understanding money is critical in the US
consumer society for which the primary means of survival is based on the ability to purchase
what is needed. Just because you were born into such a society and this society may be all you
know, does not mean that it is the only way people can live. Some cultures, both historically and
in current times, do not share our concern with the accumulation of wealth. However, since
money is important for survival in our culture, our initial focus in this book will be on financial
math, so that more prudent financial decisions can be made.
The economic recession that began at the end of 2007 and continued through 2009 has
been the first major economic shock that many of those now living have faced. Lead by
America, many parts of the world have seen virtually continuous economic growth since the
Great Depression. Being born into a continually expanding economy has allowed most to
accept, without question, that an economy will always grow and that whatever a person wants
they will be able to buy, even if they need to borrow money to be able to afford it. Mortgages
for big houses, loans for cars and college, shopping trips to the mall that can be charged to a
credit card and pay day loans for dire situations are the key to keeping the economy growing.
But the reality of the credit crunch of 2008 and the foreclosures of millions of homes may be that
building an economy on credit is not sustainable and that a new model of finances may be
required.
To help understand your financial needs and wants, use the table below to list what you own or
rent or buy with your money, what is provided for you by others (e.g. parents), what you will buy
in the future, and how much you expect it to cost.
Own/rent/ Parents
buy
provide
Will buy
in the
future
Food/water
Shelter (home/apartment)
Bicycle
Car
Recreation vehicle (boat, jet ski, RV, etc)
TV
Computer
Cell Phone
Education
Math In A Sustainable Society 2.2 – Instructors Manual
Expected
Cost
xxxxxx
6
For many of the items you listed, money is required. While most people get this money
by having a job, a very few get it through inheritance or by winning the lottery. Assuming you
are one of the multitudes that will have to work for your money, then you have three choices of
how to get what you want.
1. Save in advance and then buy when you have enough money
2. Use credit - buy now, pay later
3. Do without or reduce the size of what you want
While many might be amazed at what they can live without, even with the awareness that
they may already be living without it and that billions of other people in the world are living
without it, we often believe our happiness is connected to having certain things. With this in
mind, we will explore the mathematics of saving in advance and using credit, so that you will be
able to make more informed decisions about getting the things you want.
SAVING IN ADVANCE
There are, in general, two ways for saving money for a future purchase. One way is to
make a one time deposit of money into some sort of an investment option (stocks, bonds, mutual
funds, savings accounts, money markets accounts, certificates of deposit) and let it increase in
value until you need it. The second way is to put a little money into an investment on a regular
basis and keep doing that until you need it.
The growth of that money can occur in two ways, it can be either arithmetic growth or
geometric (exponential) growth. To understand the difference between the two, consider that you
are offered a job and then given the choice of how you would like to be paid. The job will last
for 30 days. Payment option 1 is to be paid $1000 dollars a day. Payment option 2 is to be paid
1 cent ($0.01) on the first day and then have your pay doubled every day. Which option would
you choose? The table on the next page compares the two payment options.
Math In A Sustainable Society 2.2 – Instructors Manual
7
Day Number Daily
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Option 1
Cumulative
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$1,000.00
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
1,000.00
2,000.00
3,000.00
4,000.00
5,000.00
6,000.00
7,000.00
8,000.00
9,000.00
10,000.00
11,000.00
12,000.00
13,000.00
14,000.00
15,000.00
16,000.00
17,000.00
18,000.00
19,000.00
20,000.00
21,000.00
22,000.00
23,000.00
24,000.00
25,000.00
26,000.00
27,000.00
28,000.00
29,000.00
30,000.00
Daily
Option 2
Cumulative
$
0.01
$
0.02
$
0.04
$
0.08
$
0.16
$
0.32
$
0.64
$
1.28
$
2.56
$
5.12
$
10.24
$
20.48
$
40.96
$
81.92
$
163.84
$
327.68
$
655.36
$
1,310.72
$
2,621.44
$
5,242.88
$ 10,485.76
$ 20,971.52
$ 41,943.04
$ 83,886.08
$ 167,772.16
$ 335,544.32
$ 671,088.64
$1,342,177.28
$2,684,354.56
$5,368,709.12
$
0.01
$
0.03
$
0.07
$
0.15
$
0.31
$
0.63
$
1.27
$
2.55
$
5.11
$
10.23
$
20.47
$
40.95
$
81.91
$
163.83
$
327.67
$
655.35
$
1,310.71
$
2,621.43
$
5,242.87
$
10,485.75
$
20,971.51
$
41,943.03
$
83,886.07
$ 167,772.15
$ 335,544.31
$ 671,088.63
$ 1,342,177.27
$ 2,684,354.55
$ 5,368,709.11
$10,737,418.23
Option 1 represents arithmetic growth. It doesn’t matter how much money you have,
your amount of money increases by the same amount with each time period. Option 2
represents geometric growth. The more money you have, the more money you get paid.
It is obvious that geometric growth ultimately results in the greatest increase. So what is
geometric growth?
Consider the following two sequences of numbers:
Set A: 100, 150, 200, 250 …
Set G: 100, 150, 225, 337.5 …
1000, 2000, 3000 …
0.01, 0.02, 0.04 …
Math In A Sustainable Society 2.2 – Instructors Manual
8
For each set, the original number is n0, the next number is n1, the next is n2, etc.
We will first consider the difference between two consecutive numbers in the sequence.
This difference can be shown as nt+1 – nt where t is any value in the sequence. This means that if
t = 0 then nt+1 – nt = n1– n0. If t=1 then nt+1 – nt = n2 – n1.
If we look at the difference between consecutive numbers in set A, we find the difference
is always 50; for example, 150-100 = 50 and 200 – 150 = 50.
2000 – 1000 = 1000 and 3000 – 2000 = 1000
If we look at the difference between consecutive numbers in set G, we find the difference
changes; for example, 150-100 = 50 and 225-150 = 75.
0.02 – 0.01 = 0.01 and 0.04 – 0.02 = 0.02
Because the difference between consecutive numbers in Set A is always the same, we
conclude that Set A is showing arithmetic growth, but Set G isn’t, because the differences
change.
n
Next, let’s look at the ratio t 1 of consecutive numbers.
nt
n 150
For set A, the ratio when t = 0 is 1
1.5 while the ratio when t = 1 is
n0 100
n2 200
n1 2000
n2 3000
2
1.5
1.33 .
n0 1000
n1 2000
n1 150
n1 150
1.5 while the ratio when t = 1 is
n0 100
n2 0.04
2
n1 0.02
For set G, the ratio when t = 0 is
n2 225
1.5 .
n1 150
n1 0.02
2
n0 0.01
Because the ratio between consecutive numbers is always the same in Set G, we conclude
that set G is showing geometric growth. Set A is not showing geometric growth because the
ratio changes.
From these examples we will conclude that for arithmetic growth, nt+1 – nt = a common
difference. If it is money that is growing, the common difference is the amount of interest that is
earned. Thus nt+1 – nt = I, where I is interest. With a little algebra, we can see that the amount at
time t+1 equals the amount at time t plus the interest: nt+1 = nt + I
n
For geometric growth t 1 equals a common ratio, which we will call the growth factor.
nt
n
For set G, the growth factor is 1.5. Using algebra, we can solve 1 1.5 for n1 to get n1= n0·1.5.
n0
Therefore, the amount at time t+1 equals the amount at time t times the growth factor.
n1
2 Therefore n1= n0·2
n0
Math In A Sustainable Society 2.2 – Instructors Manual
9
GROWTH RATES AND GROWTH FACTORS
Since the focus of this chapter is on money, then it is necessary to understand growth
rates in terms of money.
We are accustomed to hearing rates on TV and radio advertisements. Car companies
advertise car loan rates and mortgage companies advertise mortgage rates. These rates are
growth rates which are typically given as a percentage. Percents are difficult to use in
calculations, so the percent is typically converted to a decimal by dividing the percent by 100.
For instance, an interest rate of 6% converted into a decimal is 0.06, which is a growth rate. The
growth rate is represented with the variable r. Common names for r are the annual percentage
rate (APR) and the nominal rate.
Multiplying the annual growth rate times the amount of money invested gives the amount
of interest that is earned in 1 year. This can be shown with the formula I = P0r where P0 is the
principal that is invested and r is the annual growth rate. The total amount of money in the
account after one year is given by adding the principal that is invested and the interest that is
earned. This is shown below, along with simplification by factoring.
Principal after 1 year = Principal + Interest
P1= P0 + P0r
P1= P0(1 + r)
where, P1 is the principal in the account after 1 year,
P0 is the amount that was put into the account in the beginning (time 0),
r is the annual growth rate
1+r is the growth factor.
If the interest rate is 6%, then the growth rate is r = 0.06 and the growth factor is 1.06.
Thus, if you invest $100 in an account with 6% interest, after one year the account will have
$106.
P1 = P0(1 + r)
P1 = 100(1 + 0.06)
P1 = 100(1.06)
P1 = 106
Be aware that there are two different questions that could be asked. The first is about the
amount of interest after a given amount of time (Pt-P0), and the second is about the amount in the
account (Pt) after a given time. Most of the time the objective will be to find the amount in the
account.
FUTURE VALUE WITH ARITHMETIC GROWTH
Planning for your future can be helped by anticipating the amount of money your
investments will be worth at some future time. The future value of an investment is dependent
upon the interest rate, the time and whether growth is arithmetic or geometric.
Math In A Sustainable Society 2.2 – Instructors Manual
10
To account for times of more than 1 year, the interest formula during arithmetic growth is
changed from I = P0r to I = P0rt. The formula for the amount of money after t years would
change from P1 = P0(1 + r) to Pt = P0(1 + rt) For an investment of $100 at 6% for 3 years would
result in interest of $18 and a total value of $118 as shown below.
I = P0rt
I = 100(0.06)3
I = 18
Pt = P0(1 + rt)
P3 = 100(1 + (0.06)3)
P3 = 100(1.18)
P3 = 118
Arithmetic growth on investments occurs when the interest that is earned is removed
from the account, so that the principal always remains the same. As will soon be evident, by
keeping the interest in the account, growth can be geometric, which will ultimately result in more
money.
FUTURE VALUE WITH GEOMETRIC GROWTH
For geometric growth, it is easier to find the amount in an account first and then use this
to find the amount of interest that is earned.
If the interest rate is 6%, then the growth rate is r = 0.06 and the growth factor is 1.06.
To find a way to determine the amount of money, we will use a geometric growth model. The
original principal will be $100 so therefore P0 = $100.
The value we will get after the first time period is P1 = P0(1+r) = 100 · 1.06 = 106
The value we get after the second time period is P2 = P1(1+r) = 106 · 1.06 = 112.36: Notice
that this can also be determined by substituting P1 = P0(1+r) into the equation P2 = P1(1+r)
which will give P2 = P0(1+r)(1+r). Thus P2 = 100·1.06·1.06 = 112.36. The nice thing about
this approach is that we can find the value after the second time period by knowing only the
starting amount and the growth rate. We can simplify P2 = P0(1+r) (1+r) to P2 = P0(1+r)2.
In a similar way, P3 = P0(1+r)3 = 100·1.06·1.06·1.06 = 119.10
To be more general, Pt = P0(1+r)t Where
Pt is the value after t years
P0 is the starting value
r is the annual interest rate
t is the number of years for the investment
The formula Pt = P0(1+r)t is the simplified form of the compound interest formula. We will
modify it shortly so it can be used in more cases, but first, we will try an example.
Math In A Sustainable Society 2.2 – Instructors Manual
11
Example 1.1: Suppose you have $1000 in an account that pays 5% interest at the end of each
year. Arithmetic growth occurs because the interest is not reinvested. Geometric growth occurs
because the interest is reinvested. The table below shows a comparison of the interest earned
each year as well as the total money at the end of four years.
End of
Year
1
2
3
4
Arithmetic Growth – don’t reinvest interest
Principal
Rate
Interest
Total
Geometric Growth – reinvest interest
Principal Rate
Interest Total
$1000
$100
0.05 0.04
$50.00 $4
$1050 104
$1000 $100
0.05 0.04
$50.00 $4
$1050
$1000 $100
0.05 0.04
$50.00 $4
$1100 108
$1050
0.05 0.04
$52.50
$1102.50
$1000 $100
0.05 0.04
$50.00 $4
$1150 112
$1102.50
0.05 0.04
$55.13
$1157.63
$1000 $100
0.05 0.04
$50.00 $4
$1200 116
$1157.63
0.05 0.04
$57.88
$1215.51
Notice how much more money there is with geometric growth than with arithmetic
growth. The reason is that there is compound interest, that is, the interest is earned on the
principal and accumulated interest rather than just on the principal. Compound interest occurs
when the interest is left in the account.
THE COMPOUND INTEREST FORMULA
Interest is posted to the account at different times. Some financial institutions post it
annually; others post it quarterly or monthly, while others may post it daily or continuously. To
understand the difference between the various compounding periods, we must first determine the
number of periods in a year. We will let this value be represented by k.
Compounding period
Annual
Quarterly
Monthly
Daily
Continuously
Number of periods in a year (k)
1
4
12
365
Infinite
The interest paid in each period is equal to the APR/k. For example, if the APR is 5%
and it is compounded quarterly, then the quarterly interest rate is 0.05/4 = 0.0125.
End of
Quarter
1
2
3
4 (1 yr)
5
6
7
8 (2 yrs)
$1000 invested at 5% compounded quarterly
Principal
Rate
Interest
Total
$1000
$1012.50
$1025.16
$1037.97
$1050.95
$1064.08
$1077.38
$1090.85
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
$12.50
$12.66
$12.81
$12.97
$13.14
$13.30
$13.47
$13.64
$1012.50
$1025.16
$1037.97
$1050.95
$1064.08
$1077.38
$1090.85
$1104.49
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12
Notice that compounding quarterly results in more money than when compounding annually.
We can modify the original compound interest formula to account for more frequent
tk
r
compounding. The modified formula is Pt P0 1 Where
k
Pt is the value after t years
P0 is the starting value
r is the annual interest rate (APR)
t is the number of years for the entire investment
k is the number of compounding periods in a year
We will now use this formula on example 1.1.
Pt P0 1
r
k
tk
0.05
Pt 10001
4
2*4
Pt 1104.49
ANNUAL PERCENTAGE YIELD
As can be seen in example 1.1, after one year, we have actually increased the value of the
account by more than 5%. This is typical when interest is compounded. As already mentioned,
the interest rate that is stated for an account is called the nominal rate or the Annual Percentage
Rate (APR). The interest rate that is actually earned as a result of compounding is called the
effective rate or Annual Percentage Yield (APY). Annual Percentage Yield can be found by
calculating the rate portion of the compound interest equation for one year, then subtracting 1:
APY 1
k
r
1 . In the example of 5% compounded quarterly, we get
k
4
0.05
APY 1
1
4
APY 0.05095 .
You can verify this is the correct value by finding the actual percent increase after one year.
$1050.95 - $1000 50.95
0.05095
$1000
1000
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13
For each of the following compounding periods, find the amount of money in an account
after 2 years if the initial principal (P0) is $4000 and the interest rate is 8%. Find the APY
(rounded to 4 decimal places).
Annual Compounding
Quarterly Compounding
Monthly Compounding
tk
r
0.08
Pt P0 1 = 40001
12
k
k
2*12
= $4691.55
12
r
0.08
APY 1 1 = 1
1 = 0.0830
12
k
Daily Compounding
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14
CONTINUOUS COMPOUNDING
Did you notice the APY increases with more frequent compounding, but that the increase
is less each time? Supposing you compounded twice a day, or every minute or every second,
would there be a limit to the increase in APY? It turns out that there is a limit. This limit occurs
when you have continuous compounding. To understand this increase, we will modify the
compound interest formula (Tussy and Gustafson, 2008)
Pt P0 1
Let rn = k
tk
r
k
Since r and k are positive, then n is positive.
r
Pt P0 1
rn
1
Pt P0 1
n
trn
nrt
1 n
Pt P0 1
n
rt
n
1
As k , n and 1 2.71828 e
n
The value e is used in cases that have continuous compounding. The formula for
continuous compounding is:
Pt = P0ert Where
Pt = the value of the account after t years
P0 = the initial principal
e = 2.718282… although you should use the ex key on your calculator
r = APR
t = the number of years the money is invested
The APY when interest is compounded continuously is er-1.
Using our prior example of a $4000 investment at 8%, if the investment was compounded
continuously, the value after two years would be:
Pt = P0ert
Pt = 4000e0.08*2
Pt = 4694.04
APY = er-1
APY = e0.08-1
APY = 0.0833
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15
GRAPHING EXPONENTIAL GROWTH
In algebra you learned to graph equations on a Cartesian coordinate system graph. One
of the graphing methods was to use a table of values. We will use this method to graph the
compound interest formula equations. These are exponential equations because the independent
variable t is in the exponent.
To illustrate the graphing method, we will assume an interest rate of 3% and an initial
investment of $1000. For one graph, we will use quarterly compounding, for another, we will
use continuous compounding. To make the comparison more meaningful, we will also include a
graph for no compounding, which is arithmetic growth. Its equation is linear. The variable t
represents time, in years.
On the graph, notice that there is very little difference between the quarterly
compounding and the continuous compounding, but notice that as time goes by, the difference
between the compounding graphs and the arithmetic growth graph increases. The shape of the
graphs for which there is compounding is typical for exponential functions.
Table of values for graphing geometric and arithmetic equations using the equations:
t4
0.03
Pt 10001
, Pt = 1000e0.03*t, Pt = 1000(1 + 0.03t)
4
quarterly continuous no compounding
1000.00 1000.00 1000.00
Comparison of Quarterly, Continuous and No Compounding with a $1000
1030.34 1030.45 1030.00
investment at 3% interest
12000
1061.60 1061.84 1060.00
1093.81 1094.17 1090.00
10000
1126.99 1127.50 1120.00
1161.18 1161.83 1150.00
8000
1348.35 1349.86 1300.00
1818.04 1822.12 1600.00
6000
2451.36 2459.60 1900.00
4000
3305.28 3320.12 2200.00
4456.67 4481.69 2500.00
2000
6009.15 6049.65 2800.00
8102.43 8166.17 3100.00
0
-10
0
10
20
30
40
50
60
70
80
10,924.90 11,023.18 3400.00
Account Value ($)
t
0
1
2
3
4
5
10
20
30
40
50
60
70
80
Years
Math In A Sustainable Society 2.2 – Instructors Manual
90
Quarterly
Continuously
No Compounding
16
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Math In A Sustainable Society 2.2 – Instructors Manual
17
In-class Activity 1.1: Using the Compound Interest Formula
Name _____________________________ Effort ___/3 Attendance ___/1 Total ___/4
Pick the correct formula, show the formula, substitution and solution.
Pt P0 1
t
Pt = P0(1+r)
Pt = P0ert
r
k
APY = er-1
tk
APY 1
k
r
1
k
12t
r
d 1 1
12
Pt
r
12
Your car currently has 130,000 miles on the odometer. You are hoping it will make it to
200,000, which means it will last approximately 7 more years, based on the average amount you
drive each year. You don’t have car payments now, and would prefer not to have them in the
future. You have $2500 that you would like to invest in a 3.1% certificate of deposit,
compounded monthly. If you make this investment, how much money will be available for
buying a new car in 7 years?
r
Pt P0 1
k
tk
0.031
P7 2500 1
12
7 12
P7 3104.99
________ P7 3104.99 _____
What is the APY of this CD?
APY 1
______ APY 0.031444 ____
k
r
1
k
12
0.031
APY 1
1
12
APY 0.031444
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19
We have now seen the normal ways that financial institutions offer compounding. We
will use these methods to solve problems.
Example 1.2: An investor places $5000 in an account that pays 3.5% interest, compounded
daily. How much money will the investor have in 10 years if there are no other deposits or
withdrawals? What is the APY?
Solution 1.2: Use the compound interest formula because there is daily compounding.
Pt P0 1
r
k
tk
10*365
0.035
Pt 5000 1
365
Pt 7095.22
k
r
APY 1 1
k
0.035
APY 1
365
APY 0.03562
365
1
Example 1.3: An investor places $3000 in an account that pays 2.75% interest, compounded
continuously. How much money will the investor have in 20 years if there are no other deposits
or withdrawals? What is the APY?
Solution 1.3: Since there is continuous compounding, we need the continuous compounding
formulas.
Pt = P0ert
Pt = 3000e0.0275*20
Pt = 5199.76
APY = er-1
APY = e0.0275-1
APY = 0.02788
Math In A Sustainable Society 2.2 – Instructors Manual
20
Example 1.4: Planning ahead. Suppose the parents of a newborn want to have $20,000 in a
college fund in 18 years. How much money must they invest now, as a one-time investment, to
achieve their goal if the investment pays 4.6% interest, compounded monthly?
Solution 1.4: Because compounding is monthly, we need the compound interest formula. This
time, however, we know the value after 18 years (Pt) but we don’t know the initial value (P0).
Pt P0 1
r
k
tk
0.046
$ 20,000 P0 1
12
18*12
$ 20,000 P0 2.285
$ 8752.29 P0
Example 1.5: Planning ahead. Suppose the parents of a newborn want to have $20,000 in a
college fund in 18 years. How much money must they invest now, as a one-time investment, to
achieve their goal if the investment pays 4.6% interest, compounded continuously?
Solution 1.5: Because compounding is continuously, we need the continuous compound interest
formula. We know the value after 18 years (Pt) but we don’t know the initial value (P0).
Pt = P0ert
20,000 = P0e0.046*18
$ 20,000 P0 2.2887
$ 8752.29 P0
RULE OF 72
There is a short cut way to estimate the growth of money. It is based on continuous
compounding and will be explained without proof. Instead of calculating how much money will
be in the account after time t, the short cut approximates how long it will take to double the
initial principal. The banking industry uses the rule of 72. The rule of 72 says to divide 72 by
100*r. The result is the number of years it will take to double the initial principal. For example,
if $3000 is invested at 2.75%, then the doubling period will be 72/2.75 = 26.2 years. We can
check if this is approximately correct with the compound interest formula
Pt = P0ert
Pt = 3000e0.0275*26.2
Pt = 6166.38 – this result is slightly more than double our original investment of $3000.
To avoid confusion, please note that the Rule of 72 is the only formula in this chapter in which
the interest rate is used as a percent rather than changed into a decimal.
Math In A Sustainable Society 2.2 – Instructors Manual
21
In-Class Activity 1.2 : Using the Continuous Compound Interest Formula
Name _____________________________ Effort ___/3 Attendance ___/1 Total ___/4
Pick the correct formula, show the formula, substitution and solution.
t
Pt = P0(1+r)
Pt = P0ert
Pt P0 1
r
k
APY = er-1
tk
APY 1
k
r
1
k
12t
r
d 1 1
12
Pt
r
12
Your car currently has 130,000 miles on the odometer. You are hoping it will make it to
200,000, which means it will last approximately 7 more years, based on the average amount you
drive each year. You don’t have car payments now, and would prefer not to have them in the
future. You have $2500 that you would like to invest in a 3.1% certificate of deposit,
compounded continuously. If you make this investment, how much money will be available for
buying a new car in 7 years?
Pt = P0ert
___ Pt = 3105.86___________
0.031∙7
Pt = 2500e
Pt = 3105.86
What is the APY of this CD?
________ APY = 0.031485_______
APY = er-1
APY = e0.031-1
APY = 0.031485
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22
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Math In A Sustainable Society 2.2 – Instructors Manual
23
SINKING FUND
Up to now, we have considered one-time investments. That means we put the money
into the account and leave it there to accumulate interest until the end of the time period. Most
people do not have the necessary funds to make this type of investment. For most, the ideal is to
invest a smaller amount of money each month. This is called a sinking fund. Just like problems
with the compound interest, we would like to be able to calculate how much money we will have
after time t if we make a regular monthly deposit and also how much our regular monthly deposit
needs to be to achieve our goals. The formula for sinking fund is given without proof.
d 1
Pt
kt
r
1
k
where
r
k
Pt = the amount after time t (years)
d = the regular deposit
r = APR
k = the number of regular deposits per year
Most people make regular deposits once a month, so we will simplify the formula to
solve that type of problem.
12t
r
d 1 1
12
Pt
r
12
Example 1.6: A student decides to drink one less latte a day, thereby saving $2.50 per day. At
the end of the month, the student has saved $75.00 ($2.50*30). Every month the student puts
that money into an account that pays 4.5% interest. In 40 years when the student retires, how
much money will be in the account? How much money will the student have put into the
account?
Solution 1.6: Because we are using a regular monthly deposit, we need the sinking fund
formula.
12t
r
d 1 1
12
Pt
r
12
Math In A Sustainable Society 2.2 – Instructors Manual
24
0.045 40*12
751
1
12
P40
0.045
12
P40 $100,586.30 This is the amount in the account at the end of 40 years.
Since $75 is being deposited every month for forty years, the amount of money the
$75 12 months 40 years
$36,000 . By subtracting the
student put into the account is
month year
amount the student put into the account from the amount that was in the account after 40 years,
(100,586.30 – 36,000) we find the student earned $64,586.30.
Math In A Sustainable Society 2.2 – Instructors Manual
25
In-Class Activity 1.3: Using the Sinking Fund Formula
Name _____________________________ Effort ___/3 Attendance ___/1 Total ___/4
Pick the correct formula, show the formula, substitution and solution.
t
Pt = P0(1+r)
Pt = P0ert
Pt P0 1
r
k
APY = er-1
tk
APY 1
k
r
1
k
12t
r
d 1 1
12
Pt
r
12
Your car currently has 130,000 miles on the odometer. You are hoping it will make it to
200,000, which means it will last approximately 7 more years, based on the average amount you
drive each year. You don’t have car payments now, and would prefer not to have them in the
future. Your payment used to be $200 per month. You decide to pay that same amount each
month to an account that will pay 3.1% interest. If you make this investment, how much money
will be available for buying a new car in 7 years?
_____ Pt 18735.22 _______
12t
r
d 1 1
12
Pt
r
12
0.031 127
200 1
1
12
Pt
0.031
12
Pt 18735.22
How much of your money will you have put into the account?
_______$16,800_________
$200 12months
7 years 16,800
month 1year
How much interest would you have earned?
________1,935.22_________
18735.22-16,800 = 1,935.22
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26
This Page Is Available For Notes, Doodling, Ideas or Computations.
Math In A Sustainable Society 2.2 – Instructors Manual
27
BIG PURCHASES
Many students will eventually make big purchases in their lives. These include a home, a
car, education, or a business. They are the types of purchases that generally require a loan. We
will look at two aspects of loans, the first being to determine the monthly payment; the second is
to understand how loans are paid off.
MONTHLY PAYMENT
If the interest rate on a loan remains fixed, then the amount of the monthly payment can
be computed with the following monthly payment formula.
r
P0
12
M
where:
12t
r
1 1
12
P0 is the amount of the loan
r is the APR
t is the number of years of the loan
M is the monthly payment
Example 1.7: What is the monthly payment of a 30-year, $120,000 mortgage with a 7% interest
rate?
Solution 1.7:
r
P0
12
M
12t
r
1 1
12
0.07
120,000
12
M
1230
0.07
1 1
12
M=$798.36. This is the monthly payment.
If a person makes 360 monthly payments of $798.36, they will pay a total of $287,409.60
for the loan.
The amount of interest they pay is $287,409.60 - $120,000 = $167,409.60
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28
Example 1.8: Suppose the person had a 15-year mortgage instead of the 30-year mortgage.
How much would the monthly payment be? How much would they pay for the loan? How
much interest would they pay?
Solution 1.8:
r
P0
12
M
12t
r
1 1
12
0.07
120,000
12
M
1215
0.07
1 1
12
M=$1078.59
If a person makes 180 monthly payments of $1078.59, they will pay a total of
$194,146.20 for the loan.
The amount of interest they pay is $194,146.20 - $120,000 = $74,146.20.
Notice that a shorter loan period means an increase in the monthly payment, but a
decrease in the total amount of interest that is paid. When the house is paid in full, the person
with the 15-year mortgage would have $93,263.40 more than the person with the 30-year
mortgage ($167,409.60 - $74,146.20 = $93,263.40).
AMORTIZATION
It is important to understand what happens when you make monthly payments. Consider
example 1.7 above. After the person signs the mortgage papers, they don’t owe the bank any
money for one month. During that month, they have been borrowing $120,000. Because they
are borrowing, the loan is accumulating interest and the bank wants to be paid that interest. The
monthly payment will first be used to pay the interest, and then whatever remains will be used to
reduce the principal. The best way to see this is with an amortization table.
Payment
number
0
1
2
3
Interest
Principal
Balance
(interest/month *balance)
Payment-interest
Balance - Principal
0.07
120,000 $700
12
$798.36-700 = $98.36
$798.36-699.43=$98.93
0.07
119901.64 $699.43
12
698.85
99.51
$120,000
120,000-98.36 = $119,901.64
119,901.64-98.93 = $119,802.71
Math In A Sustainable Society 2.2 – Instructors Manual
119,703.20
29
An amortization table is constructed using 4 columns. The first column lists the months.
The second column lists the amount of interest that will be paid during the month. The third
column lists the amount of the monthly payment that will be applied to the principal and the
fourth column lists the new balance at the end of the month.
When creating a table, start with month 0 to provide a row for the original balance (the
amount that is borrowed and that must be repaid). The interest and principal columns in this row
should remain empty.
In the interest column, calculate the amount of interest that is owed for that month. When
learning about the compound interest formula you saw that the monthly interest rate was found
by dividing the annual rate by 12 since there were 12 compounding periods in a month. A
similar approach is taken to determine the amount of interest that must be paid during the month.
0.07
0.5833 .
If the annual interest rate for the loan is 7.0% then the monthly interest rate is
12
During the first month, you have been borrowing the original amount of the loan for one month.
The institution that loaned you the money would like to be paid the interest. The amount of
interest you owe at that time is the monthly interest rate times the original balance. In this
example, we find the interest that is owed for borrowing $120,000 for one month at 7% interest
0.07
120,000 700 .
rate is $700.00. This is shown by the calculation
12
Since the monthly payment of $798.36 exceeds the amount of interest, then the difference
between the two is applied to the principal thereby reducing the balance. The difference is found
by subtracting the interest from the monthly payment.$798.36 - $700.00 = 98.36.
In the last column, we see that the balance is reduced by the amount of principal that was
paid. Therefore the new balance can be found by subtracting the value in the principal column
from the balance of the previous month, 120,000-98.36 = 119,901.64.
Now the process starts all over again. For the next month, you will only be borrowing
$119,901.64 rather than $120,000. In the interest column, multiple the new balance by the
monthly interest rate. Notice that the amount of interest that must be paid is less than it was
during the first month. In the principal column, subtract the interest for the month from the
monthly payment to determine the amount that will be paid towards the principal. Notice that
the amount paid towards the principal is slightly higher than in the previous month. Finally in
the last column, find the new balance at the end of the month by subtracting the principal from
the previous balance.
THE EFFECT OF PREPAYMENT
It is possible to pay more than your monthly payment. One way of doing this is to
include the following month’s principal amount with the current month’s payment. For example,
if the first month’s check was increased by 98.93 to $897.29, then you would save yourself
$699.43, which is the amount of the interest you pay in the second month. You would not notice
this savings until the loan is paid off however. By paying the principal for one month, you
Math In A Sustainable Society 2.2 – Instructors Manual
30
would actually finish paying for your loan one month early. Thus, instead of making 360
payments, you would only need to make 359. Paying the next month’s principal in addition to
your regular payment will not allow you to skip the payment next month; it will only let you
finish paying for the loan one month early.
Prepaying the next month’s principal makes more sense early in the loan than later in the
loan. Early in the loan, the amount that is applied towards the principal is a small amount,
whereas it is much larger later in the loan. Following is the last few months for the amortization
table. Prepayment at the end of the loan would require increasing your monthly payment
$793.73 and the savings by not having to pay interest would be only $4.63. Thus, the
prepayment benefits are greater when the prepayment is made early.
Payment
number
358
359
360
Interest
Principal
Balance
(interest/month *balance)
Payment-interest
Balance - Principal
$
$
$
13.81
9.23
4.63
$784.55
$789.13
$793.73
$ 1,582.86
$
793.73
$
0.00
When borrowing money, take the time to read and understand the loan papers you are
signing. One of the conditions within the loan papers that you should identify is that
prepayments can be made at any time, without penalty. This way, you can reduce your debt
quicker without being penalized.
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31
In-Class Activity 1.4: Monthly Payments and Amortization
Name _____________________________ Effort ___/3 Attendance ___/1 Total ___/4
Pick the correct formula, show the formula, substitution and solution.
t
Pt = P0(1+r)
Pt P0 1
r
k
tk
APY 1
k
r
1
k
12t
r
d 1 1
12
Pt
r
12
r
P0
12
M
Pt = P0ert
APY = er-1
12t
r
1 1
12
Your car finally reached 200,000 miles and you decided it was time for a new car. The
new hydrogen fuel cell car you want will cost $38,000. Based on the combination of Activity 2
and 3, you have saved 3105.86 + 18,735.22 = 21,841.08. This money will be used for the down
payment but you will need a loan for the balance. The best loan rate you can find is 8.6% for a 5
year loan.
Calculate your monthly payment.
38,000 – 21,841.08 = 16,158.92
r
P0
12
M
12t
r
1 1
12
_______332.30_________
0.086
16158.92
12
= 332.30
M
125
0.086
1 1
12
Complete the first 3 months of the amortization table.
Payment number
0
Interest
r
balance
k
xxxxxxxxxxxxxxxx
Principal
Balance
Payment-interest
Balance - Principal
xxxxxxxxxxxxxxx
1
115.81
216.49
16,158.92
15,942.43
2
114.25
218.05
15,724.38
3
112.69
219.61
15,504.77
Notice how the amount of interest paid each month is gradually decreasing, the amount
of principal paid each month is gradually increasing and the balance is gradually declining.
After 60 months, the balance will be zero.
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32
CREDIT TROUBLE
As is evident during the economic downturn of 2008-2009, credit can cause problems for
people. Borrowing beyond your means or allowing debt to accumulate on credit cards puts
people into a difficult situation when jobs are cut or anticipated raises don’t materialize because
the economy is depressed. Adopting some personal rules about how you will use a credit card
can make your life less stressful. Some rules to consider about credit cards and other credit:
1. Pay the entire amount on the credit card each month, that way you won’t owe any interest
and you will build your credit score for future borrowing.
2. Pay all loans on time so you don’t incur late fees.
3. If you can’t afford a purchase, buy it later when you have saved enough money.
4. Keep a low limit on your credit card.
5. Have only one credit card and use it only for emergencies.
6. If your credit card debt is growing, pay more than the minimum amount.
7. Live simply – quality of life is not determined by how many things you have.
8. If all else fails, cut up your credit cards and contact a credit counselor.
In times of desperation, some people resort to pay-day loans. These are loans that allow
short term borrowing with expected payback periods of two weeks. For the convenience they
offer, the borrower pays a high interest rate. Payday loans will be explored in the next activity.
The interest rates that you will determine are realistic.
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33
In-Class Activity 1.5: Payday Loans
Name______________________Points ___/4 Attendance ___/1 Total ___/5
Your savings are gone, your checking account is nearly gone and you have a bill that
must be paid. What do you do? For some, the solution is payday loans. Payday loans are short
term loans that must be paid back when you get your next pay check. If they aren’t paid back,
other fees will have to be paid.
One payday loan business lends money to customers with jobs. They charge $18 for each
$100 that is loaned. The term of the loan is 14 days.
(1) 1. What is the amount of interest that must be paid for borrowing $500?
$18
$500 $90
$100
(1) 2. What is the interest rate, r? This is not the APR!
$18
18% 0.18
$100
(1) 3. Since the term is k = 14 days, then we can calculate the daily interest rate by dividing r
r
by k, . What is the daily interest rate?
k
r 0.18
=
0.0129
k 12
(1) 4. You can use the daily interest rate to determine the annual interest rate. That is done by
multiplying the daily interest rate by 365. If you used the decimal form of the interest rate, then
multiply that answer by 100 to find the interest rate as a percent. What is the annual interest rate
(as a percent)?
0.0129∙365∙100 = 469.29%
Math In A Sustainable Society 2.2 – Instructors Manual
34
This Page Is Available For Notes, Doodling, Ideas or Computations.
Math In A Sustainable Society 2.2 – Instructors Manual
35
In-Class Activity 1.6: Excel Monthly Payment and Amortization Schedule
Name________________________________ Points ______/15 Attendance ___/5 Total ___/20
The goal of this activity is to use Microsoft Excel to find the monthly payment for a loan
and then create an amortization schedule. The spreadsheet should be versatile in that you should
be able to change input variables and have the spreadsheet recalculate. An example is shown
below.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
B
C
Cost
Down payment
Balance
Interest (as a decimal)
Term (months)
=B1-B2
Monthly Pmt
=PMT(B4/12,B5,-B3,0,0)
Period
Interest
Principal
1
2
=$B$4/12*D11
=$B$4/12*D12
=$B$7-B12
=$B$7-B13
D
Balance
=B3
=D11-C12
=D12-C13
The monthly payment formula is =PMT(rate as a decimal, number of payment period, Present
Value (negative loan amount), Future Value (0), Payment at end of period (0))
You have decided to buy a house. The house will cost $169,000. You have saved
enough for a down payment of $30,000. The interest rate for the mortgage is 5.9%, regardless of
the term. You aren’t sure if you want a 15-year mortgage or a 30-year mortgage. To decide, you
must consider whether the payments are affordable. A monthly mortgage payment should be
less than 25% of your monthly income.
(1) 1.
(2) 2.
(1) 3.
(2) 4.
What is the monthly payment for the 15-year mortgage?
1,165.46
What is the total amount of interest you will pay over the life of the mortgage? 70,783.64
What is the monthly payment for the 30-year mortgage? 824.46
What is the total amount of interest you will pay over the life of the mortgage? 157,805.50
(2) 5. What is the difference in the amount you will pay in interest over the life of your
mortgage between the 30-year and 15-year mortgages? 87,021.86
(1) 6. If your monthly income is $3,800, which mortgage can you have so that your monthly
payments are less than 25% of your income?
Select all that apply by underlining: 15-year
30-year
(6) 7. Complete the table below that shows the payment number, interest, principal and balance
for the 180th payment of both the 15 and 30 year mortgages.
Payment number Interest
Principal
Balance
15 year mortgage
180
5.70
1159.76
0
30 year mortgage
180
485.12
339.34
98,329.80
Math In A Sustainable Society 2.2 – Instructors Manual
36
This Page Is Available For Notes, Doodling, Ideas or Computations.
Math In A Sustainable Society 2.2 – Instructors Manual
37
Chapter 1 Homework
Name ____________________________________ Points _______/_____
For questions 1 – 5 answer all of the following questions. How much money will the student
have at the end of 5 years? How much interest will the student have earned in 5 years? Is this an
example of arithmetic or geometric growth? What is the effective yield (APY)?
1. A student puts $1000 in a savings account that pays 2% annual interest. The interest is
paid to the customer at the end of each year and is not reinvested.
Pt = P0(1 + rt)
Arithmetic Growth
P5 = 1000(1 + 0.02∙5)
APY = 0.0200 or 2.0%
P5 = 1100
2. A student puts $1000 in a savings account that pays 2% annual interest. The interest is
reinvested.
Pt = P0(1 + r)t
Geometric
P5 = 1000(1 + 0.02)5
APY = 0.0200 or 2.0%
P5 = 1104.08
3. A student puts $1000 in a savings account that pays 2% annual interest, compounded
quarterly. The interest is reinvested.
r
Pt P0 1
k
tk
k
r
APY 1 1
k
54
4
0.02
0.02
P5 1000 1
APY 1
1
4
4
P5 = 1104.90
APY = 0.02015 or 2.015%
Geometric Growth
4. A student puts $1000 in a savings account that pays 2% annual interest, compounded
monthly. The interest is reinvested.
Pt P0 1
r
k
tk
APY 1
k
r
1
k
512
12
0.02
0.02
P5 1000 1
APY 1
1
12
12
P5 = 1105.08
APY = 0.02018 or 2.018%
Geometric Growth
5. A student puts $1000 in a savings account that pays 2% annual interest, compounded
daily. The interest is reinvested.
Pt P0 1
r
k
tk
0.02
P5 1000 1
365
P5 = 1105.17
APY 1
5365
k
r
1
k
365
0.02
APY 1
1
365
APY = 0.02020 or 2.020%
Math In A Sustainable Society 2.2 – Instructors Manual
38
6. Use the results of problems 1 to 5 to make a graph of the number of compounding
periods in a year and the APY. The number of compounding periods should go on the xaxis, the APY goes on the y-axis. Pick an appropriate scale.
7. How much money do you have to invest one time in a 6% account compounded quarterly
to have $10,000 in 4 years?
Pt P0 1
r
k
tk
0.06
10, 000t P0 1
4
$7,880.31 = P0
44
8. Suppose you have a one-year old child and want to invest some money for a college fund.
You expect to need the money in 17 years. If you want to have $20,000, how much
money will you need to put into the account if it pays 5%, compounded daily?
r
Pt P0 1
k
tk
17365
0.05
20, 000t P0 1
365
$8,548.80 = P0
9. A student puts $1000 in a savings account that pays 2% annual interest, compounded
continuously. How much money will the customer have at the end of 5 years? How
much interest will the customer have earned in 5 years? Is this an example of arithmetic
or geometric growth? What is the effective yield (APY)? Look at the graph in problem
6, does your answer make sense?
Pt = P0ert
P5 = 1000e0.02∙5
P5 = 1105.17
Geometric
APY = er-1
APY = 0.02020134
Math In A Sustainable Society 2.2 – Instructors Manual
39
10. Use the rule of 72 to estimate the time it takes to double your principal if you invest at the
following interest rates.
a. 12% 6 years
b. 8%
9 years
c. 6%
12 years
d. 4%
18 years
e. 2%
36 years
f. 1%
72 years
11. A student estimates she needs $10,000 in 10 years. If she makes a one-time deposit, how
much money must she put into an account that pays 6% annual interest, compounded
continuously? How much interest will the customer have earned in 10 years? What is
the effective yield (APY)?
Pt = P0ert
APY = er-1
0.06∙10
10,000= P0e
APY = 0.0618
$5,488.12 = P0
12. If you deposit $50 per month for the next 15 years into a 5% account that is compounded
monthly, what is the total amount of money you will have 15 years from now? How
much interest will you earn?
12t
0.05 1215
r
d 1 1
50 1
1
12
12
P15
P15 = $13,364.45
Pt
0.05
r
12
12
13. If you deposit $25 per month for the next 8 years into a 4% account that is compounded
monthly, what is the total amount of money you will have 8 years from now? How much
interest will you earn?
12t
r
d 1 1
12
Pt
r
12
0.04 128
25 1
1
12
P8
P8 = $2,822.96
0.04
12
$25 12months 8 years
Monthly investment:
$2, 400
month 1year
Difference: $2,822.96 - $2,400 = $422.96 interest
14. How much money must you deposit per month to have $13,000 in 4 years if the APR is
3%, compounded monthly?
12t
0.03 124
r
d 1 1
d 1
1
12
12
13, 000
d = $255.25
Pt
0.03
r
12
12
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40
15. You need to borrow $8,000 to start a business. The bank offers a loan rate of 11% APR
for a 6-year loan. What is your monthly payment? What is the total amount of money
you will pay over 6 years if you don’t prepay?
r
0.11
P0
8000
12
12
M
M = $152.27
M
126
12t
r
0.11
1 1
1 1
12
12
Monthly payments:
$152.27 12months 6 years
$10,963.44
month 1year
16. Complete the first 3 months of an amortization table for problem 15.
Month
X X X X X X X
1
2
3
Interest
Principal
Balance
X X X X X X X X X X X X X X X X $8000
73.33
78.94
7921.06
72.61
79.66
7841.40
71.88
80.39
7761.01
17. You borrow $14,000 for a new car. The bank offers a loan rate of 9% APR for a 5-year
loan. What is your monthly payment? What is the total amount of money you will pay
over 5 years if you don’t prepay?
r
P0
12
M
12t
r
1 1
12
0.09
14, 000
12
M = $290.62
M
125
0.09
1 1
12
$290.62 12months 5 years
Monthly payments:
$17, 437.20
month 1year
18. Complete the first 3 months of an amortization table for problem 17.
Month
X X X X X X X
1
2
3
Interest
Principal
X X X X X X X X X X X X X X X
X
105.00
185.62
103.61
187.01
102.21
188.41
Balance
$14,000
13,814.38
13,627.37
13,438.96
Math In A Sustainable Society 2.2 – Instructors Manual
41
Chapter 1.5 Sustainability
The finance math that you learned in Chapter 1 is of relevance today, but whether that
will be the case in the future may be determined by choices humans make. You should have just
completed your first project called the Quantitative Assessment of the World. In looking at the
various graphs provided by your classmates, there are a few things that you may have noticed.
1. The population of the world continues to grow (currently about 6.8 billion).
2. The world’s ability to produce oil has probably peaked or will peak within your lifetime while
demand will grow.
3. The marine fisheries, a major source of food, are declining to the point of being critical.
4. Manmade toxic chemicals are found in our air, water and food.
5. Global climate change threatens the planet.
This situation can be viewed in a least two different ways. The first way is the way with
which you are most familiar. During your lifetime, you have had a nearly endless supply of
anything you want. The food shelves at the local grocery stores have always been full with food
produced thousands of miles away. The clothing stores have always had the latest fashions, with
the clothing produced around the world. For many students, there is no memory of a time when
there wasn’t a computer, cell phones or the internet. There has always been gas for your car.
You have been told to learn as much as possible in school so you can get a job that will keep the
society going and allow you to become a consumer in it. You are taught in schools so that you
can use your knowledge to help solve any problem you encounter. With science fiction as
inspiration, we know that we can solve all of our problems and eventually find other planets to
inhabit, when there are too many on this planet.
A different way of viewing our world is from a long-term perspective. While Homo
sapiens have been around for about 200,000 years, their numbers were small, relative to the size
of the planet. The world population didn’t reach the two billion mark until the late 1920s. At
that time, the US produced about 1.7 million barrels of oil per day. Currently, with 6.8 billion
people, the world is producing about 86 million barrels of oil per day. By some estimates, the
world cannot produce more than that amount ever again and yet there will be more people with
more demand.
From the long-term perspective, consider the changes that have occurred in Washington
State. In less than 200 years, Washington State has transitioned from a state with scattered
Indian tribes and abundant natural resources to a state with about 7 million people, large homes
and a massive, congested highway system. Streams once had millions of salmon return to
spawn. Now, considerable human effort is required to keep some runs from becoming extinct.
There are more people with more needs, but with fewer resources.
While the current financial problems can be attributed to anything from corporate greed
and inadequate government oversight to too much government, consider for a moment, how
much of our economic growth is connected to the availability of cheap energy. Our ability to
extract resources, manufacture products, transport goods and people, construct buildings, grow
and transport food and heat our homes, all require energy. As the population grows, the demand
Math In A Sustainable Society 2.2 – Instructors Manual
42
for energy grows, but Earth’s store of fossil fuels declines. This connection between our energy
needs and our economic viability shows how fragile our economy is.
Consider the comments of Charles Galton Darwin (grandson of Charles Darwin) and
those of Sir Fred.
The fifth revolution will come when we have spent the stores of coal and oil that have
been accumulating in the earth during hundreds of millions of years. … It is to be hoped
that before then other sources of energy will have been developed, … but without
considering the detail [here] it is obvious that there will be a very great difference in ways
of life. … Whether a convenient substitute for the present fuels is found or not, there can
be no doubt that there will have to be a great change in ways of life. This change may
justly be called a revolution, but it differs from all the preceding ones in that there is no
likelihood of its leading to increases of population, but even perhaps to the reverse.
(Darwin, 1953, cited in Duncan, 2009).
Sir Fred Hoyle (astrophysicist)
It has often been said that, if the human species fails to make a go of it here on the Earth,
some other species will take over the running. In the sense of developing intelligence this
is not correct. We have or soon will have, exhausted the necessary physical prerequisites
so far as this planet is concerned. With coal gone, oil gone, high-grade metallic ores
gone, no species however competent can make the long climb from primitive conditions
to high-level technology. This is a one-shot affair. If we fail, this planetary system fails so
far as intelligence is concerned. The same will be true of other planetary systems. On
each of them there will be one chance, and one chance only. (Hoyle, 1964, cited in
Duncan, 2009)
Is our future the one modeled by the TV show Star Trek or is it one defined by Peak Oil
Theory and the Olduvai Theory which states that that the life expectancy of Industrial
Civilization is less than or equal to 100 years (Duncan, 2009). Can we sit back without worry
because the evidence of our lifetime is that the political leaders, scientists and engineers will
continue to make life better for us as they have done during our lifetime or should we consider
that infinite growth in a finite world is not possible and that an alternate strategy is needed for
our benefit and the benefit of our children.
The premise upon which the rest of this course will be taught is that we are capable of
developing a society that makes use of the knowledge we have gained while simultaneously
eliminating our dependence on resources that cannot be replaced. We will assume humans are
intelligent enough to continue to gain knowledge and create a world of equality and justice while
not simultaneously destroying the means for future generations to do the same thing. Such a
society would be sustainable. Sustainability means living in a way that life can be fulfilling, but
without impacting the ability of future generations to have worthwhile lives too. Sustainability
has three interrelated components- environmental, economical and social justice.
When thinking about changes to our society, keep in mind the words of Wade Davis, an
anthropologist and author of “The Wayfinder, Why Ancient Wisdom Matters in the Modern
Math In A Sustainable Society 2.2 – Instructors Manual
43
World. In a podcast available through The Long Now Foundation, Davis said “…the other
peoples of the world are not failed attempts at being us; they are unique answers to this
fundamental question. What does it mean to be human and alive?” Any alternate vision for our
future would simply be another unique answer.
To help you envision an answer that includes sustainability, we will engage in a thought
experiment. We will pretend that a group of concerned professors wish to determine how people
can live sustainably without regressing to the Stone Age. Their intent is to create an
experimental community which is isolated from the rest of civilization following its creation,
with the exception of periodic visits from the professors to gather information that can be used as
the world transitions to a sustainable society. The primary motivation for this particular thought
experiment is that the numbers we encounter in trying to understand our world are so big that
most people cannot comprehend what they mean. For example, how much is 86 million barrels
of oil per day? What enormous effort, consumption of energy and cost would it take to replace
the estimated 600 million gas powered cars that are in existence today with electric cars or
hydrogen fuel cell vehicles? While we see these numbers and know what they represent, we
cannot truly grasp them and therefore tend to ignore them.
Consequently, the community in our thought experiment will be small. It will start with
1000 people and a few assumptions.
1. There is no oil or other hydrocarbons.
2. There is no commerce outside of the community.
3. There is limited communication with those outside of the community. This means no
TVs, radios, internet.
4. People will need to produce what they need with the resources that are available.
5. What people destroy or consume will not be repaired or replenished, unless nature
does it.
6. Of the 80 square kilometers of land available, only 20% may be altered by the
residents. This will help residents understand that humans are part of the world, not the only
reason for the world to exist.
.
7. The primary objective of the community is to determine which portions of modern life
we can use sustainably and which we can do without. We will assume that people need shelter,
food and water. To be happy, they also need an opportunity to create meaning in their own life
through learning, through experiences, or by creating art or music.
For the purpose of this thought-experiment, this community will be located in the Pacific
Northwest. It will be on the coast, but will be surrounded by mountains ranging from 2500 to
4000 meters high. Its name will be Steilacoom Valley (In honor of Pierce College, Fort
Steilacoom Campus, Lakewood, WA).
Steilacoom Valley will be used for learning the mathematics of sustainable living on a
small scale, after which these skills can be applied to the large scale of our current world.
Math In A Sustainable Society 2.2 – Instructors Manual
44
Consequences Project
Having recently lived through the highest gas prices ever, the worst recession since the
Great Depression and the collapse of the housing market with lots of foreclosures, you should be
keenly aware that world events have an impact on your life. The Quantitative Assessment of the
World activity should have made you even more aware that there are a lot of issues that face all
inhabitants of this planet. You have a choice of ignoring all of this information or you can
become aware of issues and let them influence your choices.
The ability to think critically requires the formation of a question, the gathering of
information and extensive time for reflection, during which various possibilities can be mentally
explored. This project is designed to foster critical thinking that is grounded quantitatively, but
is relevant to the world as it exists today (this has nothing to do with Steilacoom Valley). The
prompt you will use is in two parts. The first part begins with the word “if” and states a
condition that you will assume to be true. The second part starts with the word “then” and
provides the topic for which you will make a hypothesis about a possible consequence. If
possible, make a hypothesis that reflects a positive vision of the future.
Example: If predictions of peak oil (now), peak coal (<10 years) and peak natural gas (<20
years) are all true, then one important consequence to the dominant culture’s worldview that
economic growth can continue forever is…
Some possible responses:
… that the dominant culture’s society will thrive and prove that growth can continue forever
… that the dominant culture’s society will collapse and prove that endless growth is a myth
… that the dominant culture’s society will be replaced by small, local, low-impact communities.
Guidelines
1. Form a group of 2 or 3 students.
2. State the complete question with your hypothesis appearing after “then”. Your hypothesis
should consist of one important expectation that you think will occur.
3. Create a systems model (optional)
4. Show relevant statistics and graphs that provide the background for the “if” portion of the
question. Identify all sources on each slide.
5. List your assumptions about the future.
6. Use original mathematical calculations to support your hypothesis. These can be formulas,
dimensional analysis, graphs that you create, etc that support your hypothesis. Do not put in
math, statistics or graphs that someone else created.
7. Conclusion – connect all the pieces (background, assumptions and math) so they support your
hypothesis.
8. Present your project publically on a tri-fold poster at a designated location on campus.
In order for you to receive feedback, create a one page outline that shows the question
and your hypothesis, the systems dynamic model (optional), the background, assumptions and
math. Submit the outline on ____________. The posters are due by _________________. The
public presentation will be on _____________.
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Topic Prompts
Oil and Energy
1. If the world’s demand for petroleum products continues at the average rate of the last decade,
but world production remains unchanged, then one important consequence to economic growth
is
2. If China’s demand for petroleum products increases at the average rate of the last decade and
their needs are met first because of political alliances with oil producing nations and if
worldwide oil production remains unchanged, then one important consequence to the US is
3. If our country stops importing petroleum over the next 10 years, then one important
consequence to transportation is
4. If petroleum and natural gas is unavailable for herbicides and fertilizers, food transport and
farm machinery, then one important consequence to the food supply is
5. If petroleum is no longer available or is very expensive, then one important consequence to
living as a result of the decrease in plastic products is
6. If our consumption of water continues at the average rate of the last decade, then one
important consequence to the world supply of fresh water is
7. If all non-essential manufacturing (toys, furnishings, new construction materials, etc) was
converted to the production of solar panels, wind mills and similar devices, then one important
consequence to our ability to live without oil is
8. If the Columbia Ice Fields melt, then one important consequence to hydro power in the Pacific
Northwest is
9. If the US eliminated all oil imports in the next ten years, then one important consequence to
the percent in poverty is
Food and Water
10. If the Columbia Ice Fields melt, then one important consequence to farming and fish in the
Pacific Northwest is
11. If the Ogallala aquifer continues to change at the rate of the last decade, then one important
consequence to food production in the affected states is
12. If ocean fishing continues at the rate of the last decade, then one important consequence to
marine fisheries is
Other
13. If the Federal Government was forced to eliminate the national debt in 10 years, using a
constant annual rate of change (so they can’t put it off until the 10th year), then one important
consequence to federal programs is
14. If the US population continues to grow at its current rate, then one important consequence to
our quality of life is
15. If gasoline powered cars were not allowed and people put half of their annual car expenses
into a mass transit system each year, then one important consequence to the ability of people to
get around is
16. If the wealth gap continues to increase in the US, then one important consequence to US
society is
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Sample Outline for Project
Question: If predictions of peak oil (now), peak coal (<10 years) and peak natural gas (<20
years) are all true, then one important consequence to the dominant culture’s worldview that
economic growth can continue forever is …
that the dominant culture’s society will be replaced by small, local, low-impact communities.
Systems Graphing
Triangles represent
wealth of population.
Fossil
Background: 1.
2.
3.
4.
5.
6.
7.
Fuels
Peak Oil
Peak Coal
Peak Natural Gas
EROEI (Energy Return on Energy Invested)
Relationship between GDP and Fossil Fuel Use
Relationship between oil costs and unemployment (lag 2 years)
Transition Communities
Assumptions: 1. National debt will lead to cuts in programs for those in poverty
2. Young people will adjust to the world situation into which they are born.
Mathematical Argument:
1. Show a regression between oil consumption and community size
Conclusion:
1. Fossil Fuels will peak in a relatively short time.
2. Communities will become no larger than can be sustained by the energy they
can obtain.
Math In A Sustainable Society 2.2 – Instructors Manual
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Evaluation
On Time: Outline (7), Poster (7) and Presentation (6)
___/20
Poster Organization and visual appeal
Excellent (9-10), Good (8) Fair (7) Poor (<7)
___/10
Speaking
___/10
Excellent (9-10), Good (8) Fair (7) Poor (<7)
History
Relevant and must contain sources
Excellent (18-20), Good (16-17) Fair (14-15) Poor (<14)
___/20
Assumptions Relevant and does not contain what was hypothesized
Excellent (9-10), Good (8) Fair (7) Poor (<7)
___/10
Math
Original and relevant.
Excellent (18-20), Good (16-17) Fair (14-15) Poor (14)
Hypothesis
___/20
Supported by background, assumptions, math and conclusion.
Excellent (9-10), Good (8) Fair (7) Poor (<7)
___/10
Total
____/100
Poster Design for Consequence Project
Left Panel
Question
Hypothesis
Systems Graphic,
(optional)
Center Panel
Title
Background Graphs
Right Panel
Assumptions
Conclusion
Math
Student Names
Provide sources for all graphs and data.
When presenting numbers in a table you make, round the numbers. For example, if you
find a population figure of 6,823,452,781, it is acceptable to round it to 6.8 billion. If you
are discussing millions or billions of dollars in money, anything less than $1000 is trivial.
The math must be of your own creation. Don’t put in graphs from another source.
All group members should be involved in answering questions
The conclusion should be consistent with the hypothesis
Your assumptions should not mention what you hypothesized
Use the math to connect the background and assumptions to the conclusion
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Systems Thinking
It is typical that the analyses of problems faced by humans are viewed in isolation.
However, topics such as climate change, population growth, peak oil, social justice and national
debt are all connected. Solving these problems requires the ability to see the connections and
understand how a change in one area can cause a change in another.
The objective of this section is to begin the process of thinking about systems. “A
system is an interconnected set of elements that is coherently organized in a way that achieves
something. … a system must consist of three kinds of things: elements, interconnections, and a
function or purpose.” (Meadows, 2008).
Examples of systems include the digestive system and circulatory system of an organism.
These are part of organisms which are a larger system. Organisms are part of larger systems
such as communities or ecosystems. Everything on this planet is part of the solar system.
The math that has been done thus far in this course and that is typically found in earlier
courses such as algebra is more algorithmic in nature, meaning that a process and rules are
followed for solving a problem. Understanding a system begins with the process of identifying
the system’s boundaries and deciding what is in the system. Ideally, the elements and the
interconnections will be shown with a graphic (picture). To produce such a graphic requires
creative thinking, brainstorming, collaboration and reflection. Developing the graphic gives your
“inner artist” an opportunity it normally doesn’t get in a math class. There will be many correct
way of making the graphic. You can include geometric shapes, arrows, draw pictures or take
another approach so that what you draw shows the system components.
The exercises we will do are meant to get you thinking about connections. Thinking this
way will make some of the math that is done in the remaining chapters more meaningful. In
Chapter 5 we will explore the idea of system dynamics modeling which will combine specific
graphic designs with algebraic formulas to show the changes in a system.
Example 1.5.1 List the components of a system that impact the energy used in a home.
Size
Shape
Window area
Window
locations
Insulation
Heating/cooling
system
Distance from
hot water tank to
faucet
Room design
Lighting
appliances
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Example 1.5.2 As a class, list the components of a system that impact a community’s
water supply.
Show the water system graphically. No examples are provided so that you can feel free
to make your own creation that will show the connections of the elements in the system.
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Some problems facing humanity are large and have many components in the system. To
help you think of system components, try to find examples from the following 5 categories:
1.
2.
3.
4.
5.
Individuals in the system
Family/Friends
Strangers
Non-human life
Natural Resources
As a class, identify components of the system that impact poverty using ideas from each
of the 5 categories.
Yourself
Family/Friends
Strangers
Non-human life
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Natural Resources
51
Activity 1.5.1: System Thinking
Name _______________________________ Points ___/4 Attendance ___/1 Total ___/5
Think of at least 5 components of a system that impact your free time then present them in a
graphic of your own design.
Think of at least 1 component from each of the 5 categories for a system that impact marine
fisheries then present them in a graphic of your own design.
Yourself
Family/Friends
Strangers
Non-human life
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Natural Resources
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This Page Is Available For Notes, Doodling, Ideas or Computations.
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Chapter 2 Population Growth
With the number of people putting a tremendous pressure on the world resources, it is
useful to understand the mathematics of population growth.
There are four different factors that affect populations. These factors are the birth rates,
the death rates, immigration and emigration. The last two are known collectively as migration.
For a completely enclosed system such as the Earth, there is no migration, so only birth and
death rates are relevant. For smaller regions, such as communities or countries, migration must
be included.
Birth rates are the number of births per 1000 individuals. Likewise, death rates are the
number of deaths per 1000 people. A population that has 20 births for every 1000 people has a
20
0.02 or 2%.
birth rate of
1000
The growth rate of any location is defined by:
Growth rate = birth rate – death rate + migration
If migration is a positive number, more people are immigrating than emigrating.
Conversely, if migration is a negative number, more people are leaving the country than arriving.
All the following rates are based on information from the 2009 CIA – World Fact Book
(CIA, 2009). Birth rates of different countries vary from a high of 51.6 per 1000 in Niger to a
low of 7.42 per 1000 in Hong Kong. The US birth rate is 13.82, while the world’s birth rate is
20.18.
Death rates vary from a high of 30.83 per 1000 in Swaziland to a low of 2.11 per 1000 in
the United Arab Emirates. The US death rate is 8.38 per 1000, while the world’s death rate is
8.23 per 1000.
Migration rates vary from a high of 22.98 in the United Arab Emirates to a low of -21.03
in Federated States of Micronesia. Migration in the United States is 4.31 per 1000, ranking it
25th in the world.
For the United States, the growth rate is 13.82 – 8.38 + 4.31 = 9.75 per 1000.
Another way to understand population growth is with the Total Fertility Rate (TFR).
This is the average number of children born to a woman during her lifetime. It is actually a
composite amount based on the expected number of births for women of different ages. TFR
vary from a high of 7.75 children per woman in Niger to a low of 0.91 in Macau, which is
similar to Hong Kong in that it is part of China, but has its own set of laws. The TFR in the
United States is 2.05 and in the world it is 2.61.
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Example 2.1: Suppose that the Steilacoom Valley population had similar birth and death rates
as the United States. How many people will be in Steilacoom Valley in 12 years?
Solution 2.1: This may, or may not be a realistic assumption, but we will make it to help explain
the change in the Valley’s population. We will also assume immigration equals emigration, so
that if anyone wants to join the Steilacoom Valley experiment, they could only do so in
replacement of someone who wants to leave.
The first difficulty encountered with trying to use the US birth and death rates is
rounding. A rate such as 13.8 births per 1000 people can make sense if there are millions of
people, but if there are only 1000 people, then there cannot be exactly 13.8 births. Consequently,
we will assume there will be either 13 or 14 births. Likewise we will assume there will be either
8 or 9 deaths. If a normal year has 14 births and 8 deaths, then there will be 6 more people
during the year. Since humans take many years to reach childbearing age, it would be a while
before the increased population would result in an increase in the number of births. On the other
hand, if the death rate stayed the same as the US, then the actual number of deaths could increase
gradually as the population increases. The table below shows the possible changes during the
first dozen years.
Year
0
1
2
3
4
5
6
7
8
9
10
11
12
Births
14
14
13
14
14
14
14
13
14
14
14
14
Deaths
8
8
8
9
9
9
9
9
9
9
9
9
Final
Population
1000
1006
1012
1018
1023
1028
1033
1038
1043
1048
1053
1059
1065
Change
Year
6
6
6
5
5
5
5
5
5
5
6
6
0
1
2
3
4
5
6
7
8
9
10
11
12
Tacoma, WA – in class example
Births
Deaths
Final
13.8/1000 8.4/1000 Population
200,000
2,760
1,680
201,080
2,775
1,689
202,166
2,790
1,698
203,258
2,805
1,707
204,355
2,820
1,717
205,459
2,835
1,726
206,568
2,851
1,735
207,684
2,866
1,745
208,805
2,882
1,754
209,933
2,897
1,763
211,066
2,913
1,773
212,206
2,928
1,783
213,352
From this chart of the first dozen years in Steilacoom Valley, we could expect a net
increase of 5 or 6 people in each year. What impact would these additional people have on the
food needs and energy needs of the community?
Example 2.2: The birth rate in the US is 13.82 per 1000 and the death rate is 8.38 per 1000,
consequently, the growth rate, ignoring migration, is 13.82 – 8.38 = 5.44 per 1000 or 0.544%.
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Change
1,080
1,086
1,092
1,098
1,104
1,109
1,115
1,121
1,128
1,134
1,140
1,146
55
(If we include migration, the growth rate is 0.975%). How many more people will be in the US
next year as a result of births and deaths?
Solution 2.2: This information is useful for determining how many new human beings are being
added to this country each year. Since the US population is approximately 310,000,000 then
5.44 extra people
1,686 ,400 extra people each year. This is roughly the equivalent
310 ,000 ,000 people
1000 people
of adding another Phoenix to the country every year. Phoenix is currently the fifth largest city in
the country. Think of all the resources that would be needed to build a city of this size each year
and all the energy needs of this many people.
Alternate Example: US population including migration
9.75 extra people
3,022 ,500 extra people approximately the size of Chicago.
310 ,000 ,000 people
1000 people
It may be somewhat easier to think in terms of the total fertility rate than the birth rate.
The relationship between these two is shown in the following graph. The linear equation y =
1.4196 + 7.06x can be used to estimate the birth rate, given the TFR.
Scatterplot of Birth Rate against TFR
TFR:Birth Rate: y = 1.4196 + 7.06*x; r = 0.9805, p = 0.0000;
r2 = 0.9613
55
50
Birth Rate (births per 1000)
45
40
35
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9
TFR
Example 2.3: If the TFR of the United States dropped to the same level as Europe, 1.50, what
would be the effect on the US population, ignoring migration?
Solution 2.3: Use the regression equation to find the birth rate.
y = 1.4196 + 7.06x
y = 1.4196 + 7.06 (1.50)
y = 12.01
The growth rate would be 12.01 – 8.38 = 3.63 per 1000.
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3.63 extra people
1,125 ,300 extra people each year. A city of this size would rank
310 ,000 ,000 people
1000 people
between Dallas, Texas and San Jose, California which are the 9th and 10th largest US cities.
From these examples, we can see that the population will continue to rise which means a
more rapid consumption of resources and space with a corresponding increase associated with
problems related to overcrowding.
MODELING POPULATION GROWTH
The population of a country or other area can be modeled mathematically. Modeling
means to find a function that can reasonably estimate the growth. Models are seldom perfect,
since trying to predict future events based upon past trends is difficult, but a good model can
make a reasonable estimate and thus be a useful planning tool.
Before we look at modeling populations, it is important to distinguish between two types
of population growth, discrete and continuous. Species that reproduce once per year exhibit
discrete growth. Such species may exhibit strange variations in population because of changes in
the environment or other prey or predators. Human populations show continuous growth in that
births are happening throughout the year. While it might be tempting to model population
growth with an exponential equation such as was used in Chapter 1, human populations, unlike
money, are subjected to the limits of the environment. Consequently, each environment has an
upper population limit called the carrying capacity. While it is possible to exceed the carrying
capacity temporarily, eventually the environment will bring the population back to its carrying
capacity. This means that for one reason or another, either the birth rate will decline or the death
rate will increase. If the birth rate is not controlled by choice, then death from starvation, disease
or conflict will keep the population at its carrying capacity. This is the basic idea behind
Malthusian Theory as described by Thomas Malthus (1766 – 1834).
A typical approach to modeling population growth for any population with continuous
growth is with the logistic function. This function reflects the idea that when the population is
low, relative to the carrying capacity, growth follows a nearly exponential model. As the
population approaches the carrying capacity, growth is reduced.
Logistic equation for continuous growth: Pt
P0 e rt
P0 e rt 1
1
K
Assumptions for the logistic formula
We assume a carrying capacity K.
The population cannot exceed K
There is no immigration or emigration.
Increasing density depresses the rate of growth instantaneously without any time lags.
The relationship between density and the rate of growth is linear.
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A sample logistic curve is shown below.
Logistic Curve for starting population of 10, carrying capactiy of 1500 and growth rate of 2%
1600
1400
1200
Population
1000
800
600
400
200
0
-200
-100
0
100
200
300
400
500
600
700
Years
Example 2.4: A town has 400 people with a carrying capacity of 1500. If the community has a
2% growth rate, how many people will they have in 10 years?
Solution 2.4: Substitute into the logistic growth equation and simplify.
P0 e rt
Pt
P e rt 1
1 0
K
400e 0.0210
=461
Pt
400 e 0.0210 1
1
1500
Logistic Growth curve with a starting population of 400 and a Carrying Capacity of 1500
1600
1400
Population
1200
1000
800
600
461
400
200
-50
10
0
100
50
200
150
300
250
400
350
450
Years
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Example 2.5: A city has a carrying capacity of 50,000 and a current population of 25,000. If
the community has a 3% growth rate, how many people will they have in 6 years?
Solution 2.5: Substitute into the logistic growth equation and simplify.
P0 e rt
P e rt 1
1 0
K
25,000e 0.03 6
Pt
= 27,244
25,000e 0.03 6 1
1
50,000
Pt
These examples are simplistic in that towns and cities are not isolated and consequently
additional food and supplies can be imported to provide additional resources for people.
However, if there is insufficient gasoline, then trade with the city will be far more restrictive and
the carrying capacity will be of more legitimate concern.
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59
In-Class Activity 2.1: Logistic Growth Activity
Name___________________________ Effort _______/3 Attendance ___/1 Total ___/4
Washington became a state in 1889. In 1900, the population in Washington was 518,000
(Caldbick 2010). Assume the carrying capacity of the state is 8.8 million, based on the criteria
used in Steilacoom Valley (only 20% of land can be used). Assume the total fertility rate had
always been the same as the US is today, 2.1 children per woman. Answer the following
question to create a graph of Washington’s population between 1900 and 2700.
If T is the Total Fertility Rate then determine B, the births per 1000, by using the regression
equation B = 7.06T + 1.4196.
Births per 1000 __16.2456_
Convert this to a birth rate rounded to 4 decimal places
Birth Rate = __0.0162_
If the death rate is always 0.0084 (current US rate), then what is the growth rate r= __0.0078_
P0 e rt
, to make a graph of the
P0 e rt 1
1
K
Complete the table of values then graph the population for each
Use the continuous form of the logistic formula, Pt
population through 2700.
year in the table.
Year
1900
2000
2100
2200
2300
2400
2500
2600
2700
Population
0
100
200
300
400
500
600
700
800
518,000
1,056,526
2,018,468
3,464,395
5,158,255
6,648,353
7,663,124
8,239,642
8,533,953
Use the logistic formula to predict Washington’s population in 2008, and then compare the
results to the actual 2008 population of 6.7 million. How do you explain the difference?
P(108) = 1,115,927. This model is based on current birth and death rates, not the rates that
actually existed since 1900. Immigration to the state was not included in this model.
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Chapter 2 Homework
Name ___________________________ Points _______/____
1a. Use the logistic formula to determine the population curve for Steilacoom Valley assuming
an initial population of 1000 and a carrying capacity of 2300. Use a growth rate of 0.54%
(0.0054) which is the US growth rate, without immigration. Complete the table of values for
each century SV exists, then graph.
0
100
200
300
400
500
600
700
800
900
1000
1000
1309
1596
1830
2001
2116
2189
2234
2261
2271
2286
1b. One thousand years is a long time for a community to exist. The United States is less than
250 years old and has seen considerable depletion of its natural resources, much of which has
occurred over the last 100 years. What do you expect will happen to the US population during
the next 750 years?
2. The most densely populated country is Bangladesh, with a population over 10 million and a
density of about 1000 per square kilometers (CIA, 2009), If we assumed this density is the
maximum for the planet, then with a total land area of about 149,000,000 km2 (Cain, 2009),
this planet could have a carrying capacity of 149 billion people. If the growth rate is 0.012 and
the current population is 6.7 billion, how many people could we expect on the planet in 100
years if the growth can be modeled logistically?
About 20.1 billion
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63
Chapter 3 The Algebra of Sustainability
For a community to survive for a thousand years or more on the available land and
resources will require different habits of living than exist in a culture that believes in endless
growth and that can maximize the consumption of resources. The creation of Steilacoom Valley
requires careful thought about what is necessary for people and what is unnecessary. Humans
have created many useful technologies to which they have become accustomed. Of those that we
may find highly desirable are a heated home, cooking facilities, heated water, laundry facilities
and medical facilities. All of these might be considered luxuries by people in various parts of the
world. Things that are not needed, although we have become accustomed to them in recent
years, are telephones, electronic products, plastics and motor vehicles.
Steilacoom Valley will be designed to provide small homes with minimal energy
requirements for all residents. Electrical energy will be provided by windmills. The population
size will be limited to the carrying capacity. The three objectives of this chapter are to
1. determine the best shape of a home so that the home will have the maximum floor area
with the minimum wall space and minimum volume of air inside to be heated
2. determine the carrying capacity
3. determine the number of windmills needed
These activities will be done in class, as a group activity.
Before calculating these, we will learn the skill of dimensional analysis. This method,
used frequently by chemists and engineers, is an ideal way to convert units.
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Dimensional Analysis Activity
Most numbers used in the real world have units. Units are words that clarify the number.
Examples of units are gallons, meters, miles, and pounds. Sometimes two units are combined
with the word “per” to give a rate. Examples of this are miles per hour and miles per gallon.
The word per indicates division so that the number of miles is divided by the number of hours to
get miles per hour.
This activity will focus on one skill that is used by chemists and engineers to make
converting from one set of units to another set easier and more organized. The skill is called unit
analysis or dimensional analysis and follows a very specific process.
Unit Conversion
We typically work with units of length, mass, volume and time or with rates such as
miles per hour or cubic meters per second. Sometimes the units given in the problem are not the
units we need, so it is necessary to convert from one set of units to the other. While it may be
easy for some to see that the conversion of yards to feet requires multiplying the quantity in
yards by 3 to get the equivalent quantity in units of feet, it is not so easy to see what must be
done to convert a rate of miles per hour into one of meters per second. The skill of dimensional
analysis makes even the most challenging conversions a simple process.
The key to unit conversions with dimensional analysis is unit fractions. Unit fractions are
fractions with different units in the numerator and denominator but in which the value of the
3 feet
numerator equals the value of the denominator. For example, the unit fractions
and
1 yard
1 yard
have different units in the numerator and denominator (feet and yard) but 3 feet equals 1
3 feet
yard. The key to using unit fractions is to recognize which units are in the numerator and which
are in the denominator.
Example 1. Convert 100 yards to feet.
Example 2. Convert 300 feet to yards
3 feet
300 feet
100 yards
1 yard
1 yard
100 yards
300 feet
3 feet
In both examples, the original value was written followed by a unit fraction. The original
value is a numerator term (with a denominator of 1). The unit fraction was written in such a way
that the units in the denominator were the same as the units of the original number, thus allowing
the units to cancel. The original number is then multiplied by all numbers in the numerator and
divided by all numbers in the denominator. Unit equivalencies are provided in the next table.
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USCS (US Customary System)
12 inches (in) = 1 foot (ft)
3 feet (ft) = 1 yard (yd)
1760 yards (yd) = 1 mile (mi)
5280 feet (ft) = 1 mile (mi)
Unit Equivalencies
USCS – Metric
Length
2.54 centimeters (cm) = 1 inch (in)
1 kilometer (km) = 0.62 miles (mi)
Metric or SI
1000 millimeters (mm) = 1 meter (m)
1000 meters (m) = 1 kilometer (km)
100 centimeters (cm) = 1 meter (m)
Area
2
1 square mile (mi )= 640 acre
1 acre = 43,560 square feet (ft2)
8 ounces (oz) = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pt) = 1 quart (qt)
4 quarts (qt) = 1 gallon (gal)
1 cubic foot (ft3)=7.481 gallons
(gal)
16 ounces (oz) = 1 pounds (lb)
2000 pounds (lb) = 1 ton
1000 Watts = 1 kilowatt
1000 calories (cal) = 1 kilocalorie
(kcal) = 1 Calorie (Cal)
1 kilowatt hour (kWh)=
3412 British Thermal Units (BTU)
1 hectare = 2.471 acre
1 square mile (mi2) = 2.59 square kilometers (km2)
Volume
1 quart (qt) = 0.946 liters (L)
Mass
2.20 pounds (lb) = 1 kilogram (kg)
1 pound (lb) = 453.6 grams (g)
Power, Energy and Work
1 calorie (cal) = 4.187 Joules (J)
1 Joule (J) = 1 Watt-Second (W·S)
1 square kilometer (km2) = 100 hectare
1 hectare = 10,000 square meters (m2)
1000 milliliters (ml) = 1 liter (L)
1000 liters (L) = 1 cubic meter (m3)
1000 milligrams (mg) – 1 gram (g)
1000 grams (g) – 1 kilogram (kg)
1000 kilograms = 1 metric ton
1 kilojoule (kJ)= 1000 joules (J)
1 megajoule (mJ) = 1,000,000 joules(J)
Time
60 Seconds (s) = 1 minute (min)
60 minutes (min) = 1 hour (h)
24 hours (h) = 1 day (d)
365 days (d) = 1 year (y)
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Dimensional Analysis
Name _____________________________ Points _____/16 Attendance ___/2 Total ___/18
Note: Use the exact equivalencies from the table and be precise in showing the process.
In problems 1, 2 and 3, the entire dimensional analysis problem has been provided; you
only need to calculate the answer. Multiply by numbers in the numerator and divide by numbers
in the denominator to obtain the converted value. Ignore the ones.
(1)
1. Lengths: In a normal home, the ceilings of a room are 8 feet above the floor. What
is this distance in meters?
12in 2.54cm 1m
hint (8·12·2.54/100)
8 ft
2.4384meters
1 ft 1in 100cm
(1)
2. Area: A home contains 2000 square feet. How many square meters is the house?
Remember to square the unit fractions.
2
12in 2.54cm 1m
2
2
2
2
2000 ft
185.8m hint(2000·12 ·2.54 /100 )
1
ft
1
in
100
cm
(1)
3. Mixed: A person consumes approximately 2000 kilocalories per day. How many
kilocalories are required by a community of 500 people for a year?
2000kcal 500 people 365days
kcal
365, 000, 000
1year
person day
year
2
2
2
In problems 4 and 5, use the Unit Equivalencies table to put in the numbers missing from
the unit fraction, then cancel units that are the same in the numerator and denominator and
multiply or divide the numbers, as appropriate.
(2)
4. Energy: Household energy consumption is calculated by multiplying the amount of
power (kilowatts) needed times the number of hours that it is used. Your electric bill is
calculated based on the number of kilowatt hours (kWh) that you use. The power requirements
of most appliances are measured in watts, while the time they are used is often measured in
minutes, thus it is necessary to convert from watt minutes to kilowatt hours. If you know the
cost of energy, you can determine how much it costs to operate an appliance.
A 1250 watt microwave oven uses 1250 watts of power. If it is turned on for 24 minutes during
the course of a day, how much energy was used in units of kilowatt hours?
1kW 1h
1250W 24 min
0.5kWh
1000W 60 min
Math In A Sustainable Society 2.2
68
(2)
5. Energy Costs: If a clothes dryer uses 4.5 kilowatts of power and the dryer is operated
for 1 hour and 10 minutes how much does it cost to dry the clothes if the cost of energy is $0.06
per kilowatt hour.
1hr $0.06
4.5kW 70 min
$0.315
60 min kWh
In problems 6 to 8, do the entire problem yourself using dimensional analysis. In all cases, show
the appropriate dimensional analysis procedure as demonstrated above and then complete the
multiplication.
(3)
this?
6. Volume: A person is supposed to drink 64 ounces of water a day. How many liters is
1c 1 pt 1qt 0.946 L
64oz
1.892 L
8oz 2c 2 pt 1qt
(3)
7. Volume: A home contains 500 cubic meters of space. What is the volume in cubic
feet? Remember to cube the unit fractions.
3
3
3
100cm 1in 1 ft
3
500m
17, 657.3 ft or
1m 2.54cm 12in
3
3
1000 L 1qt 1gal 1 ft
3
500m3
17, 662.8 ft
3
1m 0.946 L 4qt 7.481gal
(3)
8. Mixed: If a piece of land used for farming can produce 4000 kilocalories of energy
per day per acre, then how many kilojoules of energy does it produce per day per hectare?
4000kcal 1000cal 4.187 J 1kJ 2.471acre
kJ
41,384.3
day acre 1kcal 1cal 1000 J 1hectare
day hectare
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69
Determine the shape of homes in Steilacoom Valley
Name________________________Points _____/16 Attendance ___/4 Total ___/20
Homes in the United States range in size from under 100 m2 to over 400 m2. Given that
there was a time when people lived in much smaller houses, one may wonder what the smallest
sized home is that can be comfortable for a family.
(1) 1. Use dimensional analysis to determine the number of square feet in a home that has an
area of 100 square meters. The equivalencies you need are 100 cm = 1 m, 2.54 cm = 1in, 12 in =
1 ft. Be careful, because of the squared units in this problem.
2
2
2
100cm 1in 1 ft
2
100m
1076.39 ft
1m 2.54cm 12in
Assumptions for Steilacoom Valley homes:
1. Families will contain 3 or fewer people (one child at most because we are trying to limit the
population).
2. In a sustainable community, some items will be shared, so each house doesn’t need one.
Also, the amount of “stuff” a person has can be minimized.
3. Every family wants their own house, but to use less land, houses will be close to each other.
4. A home that contains the most area inside with the least amount of outer wall space is the
ideal for sustainability because less material is used to build the walls and less heat is lost
through the walls if there is less wall space.
2
Shape: What is the best shape for a home? Let’s experiment with a home that has an area of 36
square meters. The formulas we will use are A=LW, P = 2L + 2W, A = πr2, C = 2πr. Determine
the perimeter for each of the following shapes:
(1)
(1)
(1)
(1)
(1)
Rectangle: 1 x 36
Rectangle: 2 x 18
Rectangle: 3 x 12
Rectangle: 4 x 9
Square: 6 x 6
(1) Circle: r=3.385
Perimeter = 74m
Perimeter = 40m
Perimeter = 30m
Perimeter = 26m
Perimeter = 24m
Circumference = 21.27
(1) 2. What do you conclude is the best shape of a house for maximizing the area while
minimizing the distance around (perimeter or circumference)?
Circle 1: Rectangle
Square
Circle
Now consider that a house is not two dimensional (length and width) but it is three
dimensional (length, width and height). Therefore, the walls and roof, all of which require
material to build and all of which are sources of escaping energy, must be considered.
Furthermore, the volume of air inside must also be considered as larger air volumes require
greater amounts of heat. Since a square was the best rectangular shaped area, lets compare a
Math In A Sustainable Society 2.2
70
square home with 8 foot high walls and a flat roof to a round home built like a dome. This is like
half a sphere.
(1) 3. Convert the height of 8 feet to meters.
12in 2.54cm 1m
8 ft
2.4384meters
1 ft 1in 100cm
(1) 4. What is the volume of a 6 meter x 6 meter home that has an 8 foot ceiling?
V=LWH = 6m x 6m x 2.4384m = 87.78m3
(1) 5. What is the total area of the outer walls and roof?
Each wall is 6 m x 2.4384 m = 14.63 m2, the ceiling is 6 x 6 = 36 m2. There are 4 walls and 1
ceiling so 4 x 14.63 + 36 = 94.52m2.
If a dome (half sphere) is used with the radius of 3.385 m, then the volume of the dome can be
14
calculated using the formula V r 3 . The area of the outer walls can be computed using
23
1
A 4 r 2
2
(1) 6. What is the volume of the dome?
14
14
V r 3 V 3.3853 = 81.23m3
23
23
(1) 7. What is the area of the outer walls?
1
1
A 4 r 2 A 4 3.3852 = 71.99m2
2
2
8. Given that both the square and dome home have the exact same floor area, answer the
following questions to determine the better design.
(1) 8a. Which has less air to heat inside?
Square
Dome
(1) 8b. Which has less wall area through which heat is lost?
Square
Dome
(1) 8c. Which is more sustainable?
Square
Dome
This dome, which was designed by Cloud Hidden
Designs, LLC, was the winner of the Domes for the World
Design Challenge in 2008. Design parameters required
that the diameter of the single family homes must be less
than 40 feet. The cost must be less than $2,500. The
objective was to provide affordable housing in areas of the
world that suffer from poverty and natural disasters that
destroy the local homes (Kaslik, 2008, Domes of the
World Foundation, 2009).
Math In A Sustainable Society 2.2
71
Determining the Carrying Capacity of Steilacoom Valley
Part 1. Finding a function for the amount of land for homes per resident
Name____________________________Points _____/6 Attendance ___/2 Total ___/8
Show work for all problems.
Public buildings
To be a vibrant community, some public facilities are necessary. These might include a
library, community center, activity room/theater, laboratory and medical facility.
It would be very difficult to design public buildings as we don’t have enough
information, so we will just use an estimate of 20,000 m2, which is the size of the Tacoma Dome.
This should provide enough space for all public activities.
We will also estimate that workshops such as the butcher shop, bakery, furniture and
cabinetry shops, etc will occupy a facility of about 10,000 m2.
Carrying Capacity
The carrying capacity of Steilacoom Valley will be determined based on the following
assumptions.
1. Only 20% of all the land in Steilacoom Valley will be developed. The remaining land may be
used for hiking, snowshoeing or similar activities but in general it will be the amount of land that
will not be developed in anyway, ever. It will be the land that residents “allow” nature to have.
2. Food will be grown to meet the annual needs of the community, but not for export or long
term storage.
3. Land will be needed for housing, public activities, and shops
4. A safety factor of 50% will be included in the amount of land needed to account for space
between buildings and other additional area.
5. There will not be any motor vehicles or roads.
The total land area needed for SV residents is given by:
Land =1.5 (Housing + Public Building + Shops + Farmland)
This can be expressed as the combination of functions for which H(R) is a function for the land
needed for housing and F(R) is a function for the amount of farmland needed based on the
number of residents.
L(R) = 1.5 (H(R) + P + S + F(R))
Math In A Sustainable Society 2.2
72
Determine the land area needed for the homes, as a function of the number of residents.
Assumption 1: Each home will house an average of 2 people.
Assumption 2: All homes have an area of 50 m2.
(1) 1. Determine the radius of a round home with an area of 50 m2. Round your answer to the
nearest whole number.
A = πr2
50 = πr2
3.99 = r use r = 4
(1) 2. If we plan to build homes close to each other by placing them
inside a square property with the side length equal to the diameter of the
house plus 2 meters, so that houses are about 2 meters apart, then how
much land would each home require?
A = LW
= 10∙10
A = 100m2
(1)
3a. If there are 1000 residents, how much area will be required for the homes based on
our assumption of an average of 2 people per house?
2
1house 100m
2
1000residents
50,000m
2residents 1house
(1)
3b. If there are 1200 residents, how much area will be required for the homes based on
our assumption of an average of 2 people per house?
2
1house 100m
2
1200residents
60,000m
2residents 1house
(2)
3c. Generalize this by writing the function H(R) to show how much area will be required
for the homes as a function of the number of residents for any number R. H(R) should have units
of square meters. Simplify completely.
2
1house 100m
R residents
50R
2residents 1house
H(R) = 50R
Record your answer to 3c on the top of the Part 2 before turning in this activity.
If you complete this page during class, begin Part 2 of this Carrying Capacity Activity.
Math In A Sustainable Society 2.2
73
Determining the Carrying Capacity of Steilacoom Valley
Part 2. Finding a function for the amount of farmland and finding the carrying capacity.
Name____________________________Points _____/15 Attendance ___/4 Total ___/19
Show All Work
From Part 1: What is the function H(R) = 50R
V
M
Determine the amount of farmland needed for Food Production
M
M
Assumption 1: The average person in Steilacoom Valley will consume 2,500 kilocalories per
day.
Assumption 2: Grain and vegetables will be grown on a piece of land only once every four
years. During the three years it isn’t being used, it will be allowed to grow over (fallow) and can
be used by grazing animals such as bison, sheep, lamas, goats, and poultry. These animals will
be used for meat and wool, milk, eggs etc. Allowing the animals on this unused (and rotated)
farmland will result in natural fertilization. Most commercial fertilizer is produced from natural
gas, which we are assuming is no longer available.
6000kcal
Assumption 3: An estimated 6000 kilocalories can be produced per day per acre
day acre
when growing grains and vegetables. The estimate for meat is about 1200 kilocalories per day
1200kcal
per acre,
.
day acre
4.
To determine the total amount of farmland needed per person solve the two simultaneous
equations.
Equation 1: Total Calories per person per day = Vegetable/grain Calories + Meat Calories
2500 = 6000V + 1200M
where
V = number of acres for Vegetables/grain
M = number of acres for Meat
Equation 2: M = 3V since the number of acres for meat = 3 times the amount of land for
vegetables/grains.
(2)
4a. How many acres are needed per person for vegetables and grain?
2500 = 6000V + 1200(3V)
0.26 = V
(1)
4b. The total number of acres needed per person is given by N = 4V.
N = 1.04
(2)
4c. Convert the number of acres per person to square meters per person. Round to the
2
1hectare 10, 000m
2
nearest whole number. 1.04 Acres
4209m
2.471Acres 1hectare
Math In A Sustainable Society 2.2
74
(2)
4d. Generalize this by writing a function F(R) for which the amount of farmland needed
per person is a function of the number of residents. The units should be square meters.
F(R) = 4209R (or F(R) = 4215.6R if the more precise value of V=0.2604 is used)
(2)
5a. Simplify our land requirement function L(R) = 1.5 [H(R) + P + S + F(R)] by using
the generalized results from 3 and 4 to replace H(R) and F(R). Replace P and S with their
values. Combine all like terms and distribute the 1.5. Write the most simplified form of L(R).
L(R) = 1.5 [H(R) + P + S + F(R)]
L(R) = 1.5 [50R + 20,000 + 10,000 + 4209R]
L(R) = 1.5 [4259R + 30,000]
L(R) = 6388.5R + 45,000 or L(R) = 6398.4R + 45,000
The units for L(R) are square meters. This function represents the land requirement for each
resident, under the assumptions that have been made.
(2)
5b. Since land area is usually expressed in hectares, then rewrite the function by
converting the numbers to hectares. 1 hectare = 10,000 square meters
1 hectare
0.63885 hectares
6388.5 m2
2
10,000 m
1 hectare
4.5 hectares
45,000 m2
2
10
,
000
m
L(R) = 0.63885R + 4.5 or L(R) = 0.63984R + 4.5
(2)
6. The total land area of Steilacoom Valley is 80 square kilometers. Only 20% of the
land will be developed for human use. What is the largest amount of land, in hectares, that could
be developed by the settlers? 100 hectares = 1 square kilometer.
100 hectares
1600 hectares
80 km2 0.20
1 km2
(2)
7. Use the Land function for the amount of land you found in problem 6 to determine the
carrying capacity, by solving for R.
L(R) = 0.63885R + 4.5
L(R) = 0.63984R + 4.5
1600 = 0.63885R + 4.5
1600 = 0.63984R + 4.5
R = 2497.5 = 2498
R= 2493.6 = 2494
Math In A Sustainable Society 2.2
75
The Algebra of Sustainability
Energy
Name____________________________Points _____/15 Attendance ___/4 Total ___/19
At the most basic level, survival of all living organisms is dependent upon a regular
influx of energy. Most living organisms get this energy from the food they eat. It is measured in
calories. Of all the species, only one has been able to create enhanced living conditions by using
the earth’s storehouse of energy. This storehouse contains petroleum, coal, natural gas and
uranium. By using the energy stored in these resources, humans have been able to create a world
where many people can do more than simply survive. This storehouse of energy contained only
a few hundred year’s supply, given the size of the world population. It allows us to refrigerate
and cook food, heat our homes and water, wash clothes, use machinery and electronic products,
etc. The entire motivation for the Steilacoom Valley project is to relearn how to live in a world
without this stored energy, which at some point, will be insufficient. Energy, in many ways, is
the key to life.
Determine the energy requirements for Steilacoom Valley.
Power, which is the output of a generator, is measured in kilowatts, energy is measured in units
of kilowatt·hours (energy equals power multiplied by time). All the energy used in Steilacoom
Valley will be generated using windmills.
Home Energy Use
We will make certain assumptions when determining energy requirements.
Homes are small and well insulated so they only contain a small heating element.
Food is prepared in communal eating areas, not individual kitchens, but they do contain
individual small cooking appliances (burner, toaster oven).
Laundry is washed in designated areas so everyone does not need a washer/dryer
Estimated daily consumption
6 light bulbs
13 watts per bulb, 4 hours
1 tankless water heater
10 kWh/day
Space Heater
1500 watts, 1 hours
Cooking appliances
1000 watts, 0.5 hours
1. Determine the daily energy use per house in kWh. (Show work, use dimensional analysis.)
(1) Light bulbs 6bulbs
13W 1kW
1
bulb 1000 W
4hr
Water heater
(1) Space Heater
0.312 kWh
10 kWh
1kW
1500 W
1000 W
(1) Cooking appliances 1000 W
1hr
1kW
1000 W
1hr
(1) Total
1.5 kWh
0.5 kWh
12.312 kWh
Math In A Sustainable Society 2.2
76
(2)
12 .312
2. Determine the community’s daily home energy use, assuming 500 houses.
kWh 500 houses
6156 kWh
house
(2)
3. If the public buildings use 10,800 kWh per day (this is approximately the amount used
by the Pierce College Puyallup Campus), then how much energy is used by Steilacoom Valley?
6156 + 10,800 = 16,956 kWh
Assume all the electrical energy will be produced by windmills. Also, assume the
average wind speed is 18 mph. The turbines will produce a maximum power of 1000 KW of
energy with a 54 meter blade span (Layton, 2006).
30%
(American Wind Energy Association, 2009)
(2)
4. Use the Power Curve to estimate the actual turbine output for an average wind speed
of 18 mph. Show this on the graph. Change the percent to a proportion then multiply times
1000 kW. If an estimate of 30% is used then 0.30·1000kW = 300kW.
(1)
5. Multiply the turbine output times 24 hours to determine the average number of kWh
produced by each windmill in a day.
300
(2)
up.
kW 24 h
kWh
7200
windmill
windmill
6. How many of these windmills will be needed to meet the community needs? Round
16,956 kWh
2.36 Windmills 3 Windmills
kWh
7200
Windmill
(2)
7. Use a safety factor of 50% to determine how many windmills should be built. This
will allow for shut down due to problems or maintenance.
3 windmills · 1.5 = 4.5
Round up to 5. Therefore, the community needs 5 windmills.
Math In A Sustainable Society 2.2
77
Chapter 3 Homework
Name __________________________________ Points ____/___
1. Determine the carrying capacity of Steilacoom Valley using the following changes of
assumptions. All other assumptions in the carrying capacity activity will remain unchanged.
Change the average home size from 50 m2 to 70m2.
Change the average number of people per house to 3.
Change the calories per acre for grain/vegetables to 4,500 kcal/(day·acre)
Change the kilocalories per acre for meat to 1000 kcal/( day·acre)
Change the crop rotation to every 3 years, thus there are two fields for meat and one for
grain/vegetables.
1. Find radius of home
A = πr2
70 = πr2
4.72 = r use r = 5
L(R) = 1.5 [H(R) + P + S + F(R)]
L(R) = 1.5 [48R + 20,000 + 10,000 + 4670R]
L(R) = 7077R + 45,000
1600 = 0.7077R + 4.5
R = 2254.5 = 2255
2. Find area of property
A = LW
= 12∙12
A = 144 m2
3. Find function H(R).
1 house 144 m 2
48 R
R residents
3 residents 1 house
H(R) = 48R
4. Find F(R)
2500 = 4500V + 1000M and M = 2V
2500 = 4500V + 1000(2V)
2500 = 6500V
V = 0.385
Total acres per person is given by N = 3V, N = 1.154 Acres per person.
1 hectare 10,000 m 2
1.154 Acres
2.471 Acres 1 hectare
4669 .55 4670 m 2
F(R) = 4670m2
Math In A Sustainable Society 2.2
78
2. Determine the number of windmills needed for 2500 people if there is an average of 2.5
people in each home. All other assumptions in the energy activity will remain unchanged.
kWh 2500 people
12 .312
12 ,312 kWh
people
house
2 .5
house
12,312 + 10,800 = 23,112 kWh
23,112 kWh
3.21 Windmills 4 Windmills (Round up)
kWh
7200
Windmill
4 · 1.5 = 6 windmills.
Math In A Sustainable Society 2.2
79
3.
Modern society has presented us with a paradox. A century ago, if you wanted to go
somewhere relatively close to home, you either walked or used a horse. Consequently most
people were relatively fit. Since then, the invention of the automobile has allowed us to travel
greater distances in less time. We have become used to using it for even short distances.
Walking to a destination became a strange concept for many (why are you walking, don’t you
want to drive?). Of course driving, along with some of the other sedentary things we do, has led
to lower levels of fitness. Our solution for that is to drive to a favorite fitness center then
exercise on a treadmill or ride a stationary bike. While it might take longer to walk or bike to
our destination, we might be able to save a lot of time (not to mention money and resources) by
not needing to go to the gym. This problem can be used to determine which approach saves the
most time. To do so, we need to start with some basic assumptions.
Assumptions:
Assume that work is a distance of 15 miles and that because of lights, the type of roads
and congestion, the average speed is 30 miles per hour. Assume that the distance to the fitness
center is 12 miles from your home, that your average speed is also 30 miles per hour and that you
exercise for 1 hour and take an extra half hour for changing and showering. The time to work
equals the time from work. The time to the fitness center equals the time from the fitness center.
We will then compare this to bicycling to and from work every day and not using the fitness
center at all.
Organize your thoughts:
To help organize our thinking, keep in mind the things we spend time doing. We spend
time going to work, coming home from work, going to the fitness center, exercising, etc, and
returning from the fitness center. We will determine the total time involved if we use a car and if
we use a bike. You will need the formula d=rt (distance = rate·time).
Total Time = Time to work + time from work + time to fitness Center + time at fitness Center +
time to home.
Let T = Total Time
W1 = Time to work
W2 = Time from work
F1 = Time to fitness Center
F = Time at fitness Center
F2 = Time from fitness Center
T = W1 + W2 + F1 + F + F2
Math In A Sustainable Society 2.2
80
Calculations (show work):
a. Find the time to work using a car, W1.
d = rt
15 = 30t
0.5 = t (hours)
b. Find the time from work using a car, W2
W2 = W1 = 0.5 hours
c. Find the time to the fitness center, F1.
d = rt
12 = 30t
0.4 = t (hours)
d. Find the time from the fitness center, F2.
F2 = F1 = 0.4
e. Find the Total time, T when using a car.
T = W1 + W2 + F1 + F + F2
T = 0.5 + 0.5 + 0.4 + 1.5 + 0.4
T =3.3 hours or 3 hours, 18 minutes
If we chose to bicycle to work, allowed 30 minutes for a shower after we got there and bicycled
home, we would get our cardio workout and not have to use the fitness center at all, but how
much time would be involved? Assume we can bicycle at 12 miles per hour.
f. Find the time it takes to bicycle to work W1. Include a half hour to shower at work.
d = rt
15 = 12t
1.25 = t (hours)
1.25 hours to bike + 0.5 hours to shower = 1.75 hours
g. Find the time it takes to bicycle home from work W2.
d = rt
15 = 12t
1.25 = t (hours)
h. Find the Total time, T when using a bicycle.
T = W1 + W2 + F1 + F + F2
T = 1.75 + 1.25 + 0 + 0+ 0
T = 3.0 hours
i. Based on the assumptions presented in this problem, will driving or bicycling take the least
time?
Bicycling to work will take less time. Also, it will help save money on fitness club fees and car
expenses. It will also mean less contribution to pollution.
Math In A Sustainable Society 2.2
81
4. One of the hallmarks of suburbia is the grass yard. According to Adele Weder, writer for The
Tyee, an independent daily online magazine for British Columbia, a yard is “a kind of feudal
crest, marking the ability to own extravagantly useless land” (Weder 2008). For many, this part
the property demands resources such as fertilizer, weed killers and water and after using these to
help the grass grow it then requires the homeowner’s sweat and time for maintenance along with
gasoline for mowing. As the price of gas and food climbs, home owners may reconsider the
importance of a perfectly manicured lawn.
Suppose a lawn mowing service, aware that business is decreasing, decides to expand
their services by providing a service in which homeowners convert their grass yard into
productive land using permaculture. The Permaculture Institute defines permaculture as “… an
ecological design system for sustainability in all aspects of human endeavor. It teaches us how
[to] build natural homes, grow our own food, restore diminished landscapes and ecosystems,
catch rainwater, build communities and much more” (The Permaculture Institute 2007).
In Washington there is minimal rainfall in July, August and September. One of the new
services of the lawn mowing company is to calculate the anticipated water needs for the gardens
they install. They then design a system to collect and store water that lands on the roof of the
house during the rainy months. This is an alternative to letting the water run off to the streams
and the Sound and reduces the demand on city water. The objective is to determine the water
needs and the amount of rain that will be needed to store enough water.
Below is a diagram of the property of one of the company’s clients.
Shadow Area
N
Radius of home: 18 ft
The property measures 80 x 136 feet. The round home has a radius of 18 feet and the shadow
area is a trapezoid. The long side of the trapezoid is 80 feet, the short side is 36 feet and the
height is 40 feet.
Math In A Sustainable Society 2.2
82
Assumptions: All of the property except for the house and the shadow area will be planted and
will need watering. The area that is watered will need to receive one inch of water, twice a week
for 10 weeks.
Organize your thoughts:
Find the area that must be watered by finding the area of the property, then subtracting
the shadow area and half the area of the home (there is an overlap of the shadow
trapezoid and half the home).
Find the number of inches of water that must be applied.
Use the number of inches of water and the area of the home to find the number of cubic
feet of water needed.
This is the amount of water that must be stored.
How much rain must fall on the roof during the rest of the year to collect enough?
Divide the volume of water needed by the area of the roof. Convert your answer to
inches.
Calculations (show work)
a. Find the area of the property.
A=LW
A = 136 · 80
A = 10,800 ft2
( B b) h
(80 36 )h
40 2320 ft 2
b. Find the area of the trapezoid shadow A
. A
2
2
c. Find the area of the home. A = πr2
A = π182
A = 1018ft2
d. Find the area of the property that will need to be watered.
Total Property – trapezoid shadow – half of the house (other half included with the trapezoid).
10880 – 2320 – 0.5(1018) = 8051 ft2
e. Use dimensional analysis to find the number of feet of water that must be applied during the
10 week period.
2 times 1 in 1 ft
1.67 ft
10 weeks
1 week time 12 in
f. Find the volume of water that must be applied during the 10 weeks.
1.67 ft · 8051 ft2 = 13,418 ft3.
g. Find the number of inches of rain that must fall on the house to accumulate enough water.
13,418 ft 3 12 in
158 .2 in
1018 ft 2 1 ft
h. If the house is located in a place that receives an average of 35 inches of rain a year, will the
owner be able to store enough water from roof runoff?
No
Math In A Sustainable Society 2.2
83
5. In the Presidential Debate that occurred on October 15, 2008, both candidates answered the
following question asked by moderator Bob Schieffer. Would each of you give us a number, a
specific number of how much you believe we can reduce our foreign oil imports during your first
term? McCain’s answered “So I think we can easily, within seven, eight, ten years, if we put
our minds to it, we can eliminate our dependence on the places in the world that harm our
national security if we don't achieve our independence.” Obama’s answer was “I think that in
ten years, we can reduce our dependence so that we no longer have to import oil from the Middle
East or Venezuela. I think that's about a realistic timeframe.” (LA Times 2008).
To gain some appreciation for what this would mean, let’s modify the question slightly
and determine the impact of reducing oil consumption in the US to a level in which we will not
have to import any oil. That is, we will only use the oil pumped from wells in the United States.
This is not an unreasonable assumption as there will come a time when foreign countries will
want to conserve their oil resources for their own country to use and so won’t sell them to the
US. We will use the timeframe of 10 years as stated by President Obama.
The graph below shows historical US petroleum production and consumption data. It is
based on data from the Energy Information Administration website.
14
220,000,000
12
200,000,000
10
180,000,000
8
160,000,000
6
140,000,000
4
120,000,000
2
100,000,000
0
80,000,000
U.S. Field Production of Crude Oil (Million Barrels Per Day)(L)
U.S. Consumption of Crude Oil (Million Barrels Per Day)(L)
US Population(R)
Math In A Sustainable Society 2.2
US Population
240,000,000
Mar-2023
260,000,000
16
Jul-2009
18
Oct-1995
280,000,000
Feb-1982
20
Jun-1968
300,000,000
Oct-1954
22
Jan-1941
320,000,000
May-1927
24
Sep-1913
Quantity (Million Barrels Per Day)
U.S. Crude Oil Daily Production and Consumption
and US Population
84
a. To model this problem, we will need a linear equation. In January 1990, the US produced 7.5
million barrels of oil per day. In January 2000, the US was only able to produce 5.8 million
barrels of oil per day. Find the equation of the line through these two points then use the
equation to predict the amount produced in 2019 (10 years after the Obama Presidency began).
m
m
y 2 y1
x 2 x1
5.8 7.5
0.17
2000 1990
y-y1=m(x-x1)
y = -0.17x + 345.8
y-5.8=-0.17(x-2000)
y = -0.17(2019) + 345.8
y = -0.17x + 345.8
y = 2.57 million barrels/day
b. In January, 2009, the US consumed 19.1 million barrels of oil a day. To be completely off of
foreign oil in 10 years, we would have to reduce our consumption to the level you calculated in
the prior question. Find the slope of the line connecting the point (2009,19.1) to the production
amount you found for 2019.
m
y 2 y1
x 2 x1
m
2.57 19 .1
1.653 million
2019 2009
barrels per day, each year
The slope of the line you just calculated is the number of million barrels of petroleum used per
day that we would have to reduce as a nation. Assume that driving would be reduced by the
same percentage as gasoline.
c. What percent reduction would occur in the first year?
1.653
x100 8.65 %
19 .1
d. For every thousand cars on the road in 2009, how many could not be used in 2010?
0.0865 * 1000 = 86.5 or about 87 cars
e. For every hundred days you drive in 2009, approximately how many of those days could you
drive in 2010?
100 – 0.0865·100 = 91.35 or about 91.
f. What is the percent reduction that would occur after 10 years?
10 · 8.65% = 86.5%
g. For every thousand cars on the road in 2009, how many could not be used in 2019?
0.865 · 1000 = 865 cars could not be driven.
h. If this reduction occurred, how many years would it be before there would be no need for
traffic reports on the news?
i. If this reduction occurred, how would Washington State pay for the new Narrows Bridge, the
Alaskan Way Viaduct and the 520 Bridge?
Math In A Sustainable Society 2.2
85
Chapter 4 Statistics
If all of the energy needs of Steilacoom Valley are met with windmills, then it is critical
that there are sufficient winds with which to spin the turbines. This typically requires a
minimum of 10 mph winds. There are two questions that are of primary interest.
1. What is the average wind speed?
2. What proportion of time is the wind speed less than 10 mph?
The first question allows for an estimate of the possible power output of the windmills
and consequently the energy available for use in the community. The second question
determines the amount of time that energy is not being produced which can be used to estimate
the storage capacity needed.
These two questions illustrate the need for a different type of math than algebra.
Algebraic math is deterministic and as such assumes that the relationship defined by the equation
is always true. Winds on the other hand are highly variable and because of this it isn’t possible
to write an equation to answer either question directly. The only way to arrive at a definitive
answer is by recording every wind speed for every instant of time at each turbine and then
calculating either the average wind speed or the proportion of time the wind speeds were too
slow.
Our goal in this unit is to understand variables that change. These variables are called
random variables. Examples of random variables are wind speed and whether the wind speed
is over 10 mph (yes or no). We use statistics to understand these variables so that we can make
informed decisions.
Our interest is always in the population of these random variables. The population is the
entire collection of the random variables. In the case of wind speeds, we want to understand all
wind speeds that might affect the windmills.
To determine the average or proportion of the population would require a census. A
census requires getting data from every unit or person in a population and is typically not
possible because of time, money, the destructive nature of gathering data. Sometimes it is simply
impossible to gather all the data.
If we assume we cannot do a census, then the best we can do is take a sample. A sample
is a small collection of data taken from the population. A sample can only give us insight into
the population from which it is drawn, but not from any other population. For example, wind
measurements taken in winter do not give information about summer winds.
The concepts presented thus far suggest the purpose of statistics. Statistics are typically
used when there is a problem or a question about a random variable that people want to
understand in order to make a good decision. The decision often involves money or health or
quality of life issues. To make the best decision, a person would like to know the details of the
Math In A Sustainable Society 2.2
86
population, but the best a person can usually do is take a sample. The judgment about the entire
population must be based on the results of the sample.
While it is possible to determine the average or proportion of our sample data, it is highly
unlikely that this average or proportion is the same as exists in the population. This is a critical
point and the first indication that you will need a different thinking process when attempting to
understand random variables as compared to algebraic variables with which you are more
accustomed. For example, the algebraic equation 3x = 12 will give a solution of x = 4 for
everyone. However, if the average of a population is 25 and everyone in a class took a sample of
the same size from this population and found the average of their sample, in many cases, the
averages of the samples would not be 25 and most students would not have the same average as
other students.
To illustrate this concept, consider the 9 employees of a small company to be the entire
population. The data of interest is their annual salaries in units of $1000. The salaries are 15,
20, 20, 25, 25, 25, 30, 30, and 35. The only way to actually know all the values in a population
is by doing a census. As has already been discussed, doing a census is not typically possible.
However, to help explain the concept, it is useful to know the values of a very small population.
It is easy to determine that the average (mean) of this population is 25. Suppose, as is typically
the case, you did not know the mean and decided to take a sample, with replacement, of size
three from the population. In that case, there are 729 possible samples you could select and in
this case there are 13 different possible sample means you could get. The distribution of these
sample means is shown in Figure 4.1. Notice that there is only a 19% chance that the sample
mean you select would exactly equal the mean of the population. The numbers above the bar
show the number and percentage of times each sample mean would occur.
Figure 4.1
Distribution Of All Of The 729 Possible Sample Means When 3 Units Are Selected, With Replacement,
From The Population Of 15,20,20,25,25,25,30,30,35
160
141, 19%
140
126, 17%
126, 17%
120
90, 12%
90, 12%
80
60
50, 7%
50, 7%
40
Sample Means
Math In A Sustainable Society 2.2
33.3333
31.6667
30.0000
28.3333
26.6667
25.0000
23.3333
21.6667
6, 1%
20.0000
6, 1%
18.3333
1, 0%
16.6667
0
21, 3%
1, 0%
35.0000
21, 3%
20
15.0000
No of obs
100
87
Likewise, if we flip a coin ten times, we will not always get exactly 5 tails. While the
proportion of tails we should get in the long run is 0.5, we can see in the graph below that if we
flip a coin ten times and the proportion of tails varies from 0 to 1 and will equal 0.5 only 25% of
the time. Thus, the other times the sample proportion does not equal the population proportion.
Figure 4.2
Distribution Of The Proportion Of Tails If A Coin Is Flipped 10 Times
280
260
252, 25%
240
220
210, 21%
210, 21%
200
No of obs
180
160
140
120, 12%
120
120, 12%
100
80
60
45, 4%
45, 4%
40
20
0
10, 1%
10, 1%
1, 0%
0.0
1, 0%
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Proportion of Tails
Ultimately, our need for this branch of mathematics occurs when we have a question
about a population for which we want to know the average or proportion. Since we cannot do a
census, then we need to devise a way to use our sample to estimate that average or proportion.
This will require the researcher to:
1. Formulate a specific question.
2. Design a study or experiment
3. Use random sampling and good data collection methods
4. Graph the data
5. Determine relevant numerical summaries of the sample. These are called statistics.
6. Use these statistics to estimate the parameter without bias.
A parameter is a number such as a mean or proportion that summarizes all the data in a
population. A statistic is a number such as a mean or proportion that summarizes all the data in a
sample. In most populations, the statistics tend to occur randomly above and below the
parameter as can be seen in Figures 4.1 and 4.2. If for some reason, all possible statistics we
could get occur to one side of the parameter and not the other, there would be bias. This might
occur if we only sampled wind measurements on windy days.
Before sampling, it is helpful to first identify the type of data that will be sampled. Data
is either quantitative or qualitative. Quantitative data is the result of measurements or counts
Math In A Sustainable Society 2.2
88
(quantities). Examples include wind speed, exam scores, number of traffic lights on the way to
school, and how high you can jump. Qualitative data, also called categorical data, is not
numeric. Examples include heads/tails, yes/no, male/female, and true/false.
For qualitative data, the parameter or statistic of interest is the proportion. The proportion
x
of the population, which is the parameter, is p . The proportion of the sample, which is the
N
x
statistic, is pˆ . For quantitative data, the parameter or statistic of interest is the mean. The
n
x
mean of the population is symbolized with the Greek letter mu and is shown as
. The
N
x
statistic, which is the mean of the sample, is symbolized with x-bar and is shown as x
.
n
Sample size is represented with an n, while the population size, which is generally not
known, is represent with an N.
EXPERIMENTS AND STUDIES
Since gathering data usually involves considerable time and expense, it is beneficial to
have a plan of action for collecting useful data. This plan consists of conducting an
observational study or an experiment. In an observational study, units are observed and data is
recorded. In an experiment, the researcher imposes a treatment on the unit, with the intent of
determining if the treatment has a particular effect. For example, a researcher wanting to know
about the water quality of a stream would do an observational study by taking water samples
from the stream and chemically analyze them. A researcher wanting to test the effectiveness of a
fertilizer would conduct an experiment by putting the fertilizer on some of the crops and not
putting the fertilizer on the remainder of the crops. The crops that do not get fertilizer are called
the control. The purpose of a control is to provide a contrast to the units that receive the
treatment.
Observational studies would be of use to determine such things as wind speeds, energy
consumption per house, or number of kilocalories produced per acre. Wind speed data would be
collected at randomly selected times. Daily home energy use would be determined on randomly
selected homes for randomly selected days. The number of kilocalories produced on randomly
selected acres could be measured.
In cases where the researcher wants to see the effect of one random variable on another, it
may be helpful to conduct an experiment. For example, in Steilacoom Valley, conservation of
water and electricity would be important. A researcher could conduct an experiment to
determine if the use of a shower timer will help reduce the length of showers. A shower timer
can be used to indicate when 5 minutes has passed. A researcher might have some randomly
selected residents use a shower timer and others not use one. All the people in the experiment
would keep track of the length of their showers. In this example, shower timers would be the
explanatory variable or factor and the length of the showers would be the response variable.
Math In A Sustainable Society 2.2
89
The average length of a shower would be the parameter of interest. The levels are the possible
outcomes for the explanatory variable which are to use or not use a shower timer. This can be
shown in a one-way design layout table.
Factor:
Level 1
Level 2
Level 3
Response Variable
Parameter of interest
shower timer
use
not use
length of a shower
average
This experiment could be enhanced by adding a second explanatory factor. This one
could be about the shower flow rate and whether it has a high flow rate or a low flow rate. To
show this, we can use a two-way design layout table.
Response Variable: length of shower
use
Parameter of Interest: average
Factor 2:
high flow rate
shower flow rate
low flow rate
Factor1: shower timer
not use
Treatment 1
Treatment 3
Treatment 2
Treatment 4
Each treatment represents a different combination of shower timer and shower flow rate.
Thus treatment 1 represents using a shower timer with a high flow rate shower while treatment 2
represents not using a shower timer but having a high flow rate shower, etc.
SAMPLING
The key to being able to use a sample for an unbiased estimate of a parameter is random
sampling. We will learn 2 good sampling methods and discuss 2 questionable sampling
methods. The general idea that differentiates a good from bad method is that the sampler does
not make a choice, but leaves the selection up to a random process beyond his control.
One good sampling method is called simple random sampling (SRS). In its most basic
form, this means pulling names from a hat. With larger populations, numbers are assigned to
each unit in the population and a random process is used to determine which numbers are picked.
In SRS sampling, every unit has an equal chance of being selected as does every sample of size
n. The best way to pick these numbers is at random.org. An alternative is to use a graphing
calculator or a table of random digits.
Because graphing calculators are not required for this course, we will focus on using a
table of random digits to select our sample. An example is provided in Figure 4.3.
Math In A Sustainable Society 2.2
90
Figure 4.3. Table of Random Digits.
Row
Number
1
2
3
4
5
6
7
8
9
10
83984
78425
96268
58037
52354
65936
01849
94368
24504
13283
22116
65082
62423
43470
04992
11549
40765
20871
75557
33042
01657
07792
63347
88497
47754
15979
97487
13867
58840
69362
83717
43850
09111
98909
31246
92704
56378
61232
99065
92759
24799
22134
12079
79230
36779
42288
80291
87091
49850
81354
00515
76033
58082
36845
27029
07121
40351
67621
55957
76328
37723
87273
88984
30325
88187
54938
95246
27560
14117
76438
23445
13972
76565
82655
19275
08990
58004
81197
62890
29699
02705
58089
62765
48666
89632
00190
56115
63987
24961
86996
26127
12538
35923
55431
21684
81402
53197
01118
54550
65089
Think of this table as an endless string of digits between 0 and 9. The numbers are
grouped only for visual convenience.
To use the table, determine the size of the population from which you will sample.
Assign a number to each unit in the population, starting at 1 and continuing until all units have
been numbered. Count the number of digits in the unit with the highest number. This count will
be the number of adjacent digits you select.
For example, if there are 550 units in a population and you wish to select 5 of them,
starting in row 7 (an arbitrary choice), then you will look at each consecutive group of 3 numbers
(because 550 is a 3 digit number) and use them if they are less than or equal to 550. If you reach
the end of the row before you have all the numbers that you need, continue onto the next row
without skipping any numbers. Numbers with less than three digits will become three digit
numbers by putting zeros in front of them (1 will be 001, 37 will be 037).
7
8
01849
94368
40765
20871
97487
13867
56378
61232
80291
87091
40351
67621
95246
27560
58004
81197
56115
63987
53197
01118
The five numbers that would be selected are 018, 494, 076, 487 and 029.
When there are some distinctive subgroups in a population, such that the variation
between the subgroups may be more significant than the variation within each subgroup, then
stratified sampling should be used. For example, wind speeds might vary seasonally so
sampling should be done in each season. Other strata (subgroups) for other questions include
gender, age groups, locations etc. Once the strata have been determined, sampling should be
random within each stratum.
Two bad sampling techniques are voluntary and convenience sampling. Voluntary
samples give the respondent a choice to participate. Examples include web surveys and texting
in response to TV surveys. Convenience sampling is sampling those within easy access. This is
not necessarily bad, but often is a problem because those within easy access do not necessarily
reflect a good cross section of the population.
Math In A Sustainable Society 2.2
91
In-Class Activity 4.1: Simple Random Sampling and Stratified Sampling
Name _________________________Points ____/10 Attendance ____/3 Total ____/13
In the Algebra of Sustainability for Energy, we found that the average SV home will use
about 17 kWh of energy per day. One might expect considerable variation for each household
and each time of the year however. A survey will be conduction on randomly selected homes to
determine the amount of energy they use on one particular winter day. There are a total of 500
homes, so assume homes are assigned numbers from 001 to 500. Use the table of random digits,
beginning in row 2, to determine the number of the first 5 different homes to be selected.
Sampling will be done without replacement.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
46264
30267
53925
31696
74578
28015
94305
75327
08040
86958
06218
03544
51194
56714
78489
32091
00752
82717
36872
61739
40104
86651
09051
81575
86989
60523
91470
95465
40445
74577
27198
21447
30624
64038
62533
53866
13863
24107
67056
22959
62454
85801
27012
10056
44498
73148
80735
02530
76640
91486
42539
22252
60991
54602
21052
12648
43137
32039
99165
64821
04874
87969
62551
94587
59168
43919
49061
38920
81188
99892
11519
32837
71778
07145
03151
58644
76789
66705
73423
88193
43761
94780
48307
45168
46812
52432
02861
87659
53845
86719
93614
30523
99934
90109
71401
22827
76428
31020
81358
13539
53866
48478
89610
52099
40226
46019
30607
49129
20344
26497
13016
85918
03136
75447
15827
01960
40317
36765
50517
97828
83845
57935
22176
31437
02453
57335
42640
06341
77163
62380
15202
67978
41273
25781
42642
20318
61597
33712
17966
85531
35296
08526
46056
01877
74083
77461
39445
05849
11998
92557
19881
55128
58993
85685
54841
36037
49645
29071
53787
94792
(2) Simple Random Sample, first 5 homes selected _302_, __185_, __478__, __048__, __350__
Stratified Sampling. Assume the 500 homes are actually divided into 4 separate locations, with
125 homes in each location, each location named after a direction (North, South, East, West).
You would like to select 3 homes from each. Use row 4 to select from the North, row 8 from the
South, row 12 from the East and row 16 from the West. The energy use data for each home is
found on the next page. List the data associated with each randomly selected home.
(2) North _056_, _099_, _124_
Energy use data __10__, __15__, __18__
(2) South _063__, _040_, _081__
Energy use data __16__, __10__, __11__
(2) East _035_, _052_, _099__
Energy use data __15__, __17__, __10__
(2) West _031__, _107__, __020_
Energy use data __12__, __19__, __17__
Math In A Sustainable Society 2.2
92
Energy Data (kWh)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
North South East West
13
18 21
16
14
10 22
23
25
13 18
12
25
14 16
17
27
15 17
10
20
16 22
19
22
19 10
14
22
18 10
22
15
13 12
14
18
11 23
17
18
10 12
19
27
19 21
14
28
11 19
10
15
18 14
11
27
18 17
13
26
10 11
17
10
15 15
24
17
11 19
21
29
17 14
21
18
19 18
17
10
19 15
24
23
16 14
22
28
18 14
21
21
11 10
23
24
13 11
13
25
19 20
13
28
13 23
22
11
10 16
19
24
17 23
24
26
12 18
20
17
10 23
12
15
18 23
21
18
14 20
18
17
14 22
12
13
12 15
17
22
10 17
16
17
19 24
10
21
16 19
10
11
18 11
14
26
10 20
22
20
13 24
14
14
18 23
24
19
16 14
24
26
17 14
17
10
19 22
24
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
North South East West
24
17 19
14
17
13 20
22
22
17 23
15
16
12 24
16
17
12 15
15
26
19 10
24
14
15 17
22
29
16 10
24
29
12 22
13
21
12 13
16
10
13 21
22
28
14 13
16
28
14 10
14
19
11 13
19
10
11 20
14
13
10 21
20
25
12 24
13
19
16 19
11
26
17 22
17
28
18 17
10
23
10 22
11
19
16 15
12
10
17 10
24
25
12 11
17
20
12 17
15
29
12 13
10
11
14 23
22
21
18 19
15
21
12 10
17
29
19 17
14
23
13 14
18
25
18 23
24
19
17 23
13
14
17 14
20
16
18 13
23
12
11 21
23
18
15 24
22
28
16 12
21
13
15 23
21
18
12 18
19
15
12 22
22
22
19 19
16
19
11 14
17
22
10 12
21
12
16 20
15
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
Math In A Sustainable Society 2.2
North South East West
19
17 24
24
22
16 13
12
11
19 13
17
10
12 20
23
20
13 20
11
12
18 22
14
18
11 20
12
25
11 20
15
15
12 10
22
14
10 16
10
19
13 21
21
21
10 19
15
14
10 20
13
18
14 23
15
22
10 17
10
15
18 19
14
20
11 11
19
22
14 14
12
19
15 23
19
19
17 11
16
16
16 23
14
17
11 18
24
13
12 12
13
26
16 10
16
25
18 10
23
26
12 21
15
24
19 18
24
14
13 15
23
14
17 24
16
11
15 18
17
27
16 16
13
21
18 12
22
10
15 17
16
18
10 14
11
10
11 10
10
93
PROBABILITY
Since good sampling requires a random process, then it is only by chance that particular
units or subjects become part of a sample. A different random selection would result in a
different sample set of data. While this is desirable, it does mean that an understanding of basic
probability will be required to understand the theory that allows a statistic to be used to estimate
a parameter. There are three particular aspects of probability that are essential for understanding
the theory behind inferential statistics, which is the process of using statistics to estimate
parameters.
1. Simple probability – the probability of one outcome when one selection is made.
2. Or – the probability of one outcome or another outcome when one selection is made.
3. And – the probability of an outcome when more than one selection is made.
SIMPLE PROBABILITY
Probability is the proportion of times an outcome will occur over the long run. The
emphasis on long term is very important.
number of favorable outcomes
. Assume all
number of possible outcomes
possible ways are equally likely. Probability is always a numerical value between 0 and 1. This
can be shown as 0 ≤ P(x) ≤ 1. The probability is 0 if the event cannot occur. The probability is 1
if the event is a sure thing – it occurs every time.
We view probability as a fraction: P( x)
Our objective is to find probabilities of random processes. A random process (such as
random sampling) is a repeatable process whose set of possible outcomes is known, but the exact
outcome cannot be predicted with certainty. The set of possible outcomes is a sample space. A
subset of the sample space is called an event.
Example: The sample space when flipping one coin is {head, tail}. An event is getting a tail.
Example: The sample space when flipping three coins is HHH, HHT, HTH, THH, TTH, THT,
HTT, and TTT. An event is getting 2 heads.
Example 4.1: Find the probability of getting a head when flipping a coin one time.
Solution 4.1: Since there are two possible outcomes in the sample space {H,T} of which only
one is favorable {H}, then
1
P( H ) .
2
Example 4.2: Find the probability of getting two heads when a coin is flipped three times.
Solution 4.2:. Since there are eight possibilities { HHH, HHT, HTH, THH, TTH, THT, HTT,
and TTT }, of which three have two heads { HHT, HTH, THH }, then
3
P (2 heads ) .
8
Math In A Sustainable Society 2.2
94
P(A OR B)
If you are collecting data on whether students thought the economy, the environment or
social justice was the most important component of sustainability, then a simple probability
question would be to determine the probability that one randomly selected person favored the
economy.
A more challenging question is to determine the probability that one randomly selected
person favored the economy or the environment. In this case, one person is selected but there are
two possible responses that would be considered favorable. We ask the question as P(Economy
or Environment) or more generally as P(a or b). The probability is found by adding the simple
probabilities of each outcome. Thus, P(a or b) = P(a) + P(b).
Example 4.3: A school has 4000 students and 1500 believe the economy is most important,
1400 believe the environment is most important and 1100 believe social justice is most
important.
A. Find the probability that one randomly selected student thinks the economy is most
important.
number of favorable outcomes 1500
0.375
Solution A: P( x)
=
number of possible outcomes 4000
B. Find the probability that one randomly selected student thinks the economy or the
environment is most important.
number of favorable outcomes 1500 1400 2900
0.725 or find it
Solution B: P( x)
=
4000
4000
number of possible outcomes
using the P(a or b) rule:
1500 1400 2900
0.725 .
P(Economy or Environment) = P(Economy) + P(Environment) =
4000 4000 4000
Our use of the P(A or B) rule is limited to mutually exclusive events, which are events
that cannot both happen at the same time. A particularly important application of this rule is
with complements. Complements occur when there are only two possible outcomes. The
probability of one of the possible outcomes is equal to one minus the probability of the other
outcome. For example, what is the probability that a shopper will remember or not remember to
take the reusable bags into the grocery store? This can be shown as
P(remember or not remember) = P(remember) + P(not remember).
Since it must happen that the person will remember or not, then
P(remember or not remember) = 1.
Consequently 1 = P(remember) + P(not remember).
Solving algebraically for P(remember) we get P(remember) = 1 – P(not remember). Solving
algebraically for P(not remember) we get P(not remember) = 1 – P(remember).
Math In A Sustainable Society 2.2
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The Complement Rule: If A and A (called A complement) are the only possible outcomes and
they are mutually exclusive then P(A) = 1- P( A ) and P( A ) = 1- P(A).
Example 4.4: If there is a 40% chance that a shopper will forget to take the reusable bags into
the store, what is the probability that the shopper will remember to take them in?
Solution 4.4: P(remember) = 1 – P(not remember)
P(remember) = 1 – 0.4
P(remember) = 0.6
P(A AND B)
Determining simple probabilities for one selection is necessary for being able to
determine the probabilities of more than one selection. When there is more than one selection,
we are interested in finding the probability of an outcome A on the first selection and an outcome
B on the second selection etc. This is shown as P(A and B). Although sometimes this
probability can be found using simple probabilities and sample spaces, more often it is useful to
use the formula P(A and B) = P(A)P(B). This formula shows that the probability of outcome A
on the first selection and outcome B on the second selection is equal to the product of their
probabilities. We will assume that the events are independent, which means the first selection
does not affect the probability of the second selection.
Example 4.5: If a coin is flipped 2 times, what is the probability of getting 2 heads?
Solution 4.5: This can be solved in two ways.
The first way is to create a sample space and then determine the probability. The sample
1
space is {HH, HT, TH, TT}. From this, we can see the probability of two heads is .
4
The second way is to use the rule for the probability of two events. In context, we can
say that the probability of getting a head on the first flip and a head on the second flip is
1 1 1
P(H1 and H2) = P(H1)P(H2) = .
2 2 4
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USING DATA TO ANSWER QUESTIONS
The basic concepts for answering questions about a population have now been provided.
These concepts include the design of studies and experiments, the random selection process,
probability and the awareness that samples that could be selected from a population vary,
implying that many different values could be obtained for the statistic, although there will always
be a fixed, but unknown value for the parameter.
The processes for answering the researcher’s questions will be provided for both
quantitative and qualitative (categorical) data. The processes are similar, although the theory
that justifies each is different. The processes will be presented in the following sequence:
1. Graphing
2. Statistics
3. Theory for making Inferences
4. Confidence Interval
Inferences are made when sample data are used to infer something about the population.
In this text, the only inferences to be made will be with confidence intervals.
GRAPHING QUANTITATIVE DATA
Data that has been collected is typically a chaotic collection of numbers or words that at
first glance has no meaning to anyone. Consequently, the statistician needs to organize the data.
The first way to organize the data is by graphing it. This allows the researcher to see how the
data is distributed. Graphing gives a good visual impression of what the data suggests about the
population from which it was drawn.
The type of graph we use depends upon the type of data. Pie charts are used for
qualitative data while histograms are one of the primary means for graphing quantitative data.
When data is quantitative, generally a collection of measurements, then it can be graphed
with a histogram. A histogram is a bar graph in which similar size measurements are grouped
and counted. The x axis provides the lower and upper boundaries of each class (group) while the
height of the bar indicates how many values are in each class.
Steps to make a histogram.
1. Determine the lowest and highest values.
2. Create reader-friendly class boundaries
a. pick a good starting point that either equals the lowest value or is less than the
lowest value.
b. pick a class width (difference between consecutive lower boundaries) that will
produce 4 to 10 classes.
c. Show the boundaries using interval notation, for example [10,15) which would
indicate all numbers greater than or equal to 10, but less than 15.
3. Create a frequency distribution and count the number of values in each class.
Math In A Sustainable Society 2.2
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4. Make the histogram. Label the x axis with the lower boundaries. Label the y axis
with the counts. Include a graph title and axis titles on both axes.
5. Look closely at the graph to determine what the graph suggests about the data, in
relation to the question that prompted the research.
A note about reader-friendly intervals: A graph is a form of communication. If you take
all the time and effort and money to conduct important research, then it should be important for
you to communicate the results clearly. A starting value of 10, with a class width of 5, leads to
reader-friendly numbers on the x axis such as 10, 15, 20, 25, etc. On the other hand, using a
starting value of 10.2 with a class width of 4 would result in x axis numbers (10.2, 14.2, 18.2 etc)
that are definitely not reader-friendly and would make the reader have to work too hard to
understand the data. In general, the starting value should be a multiple of the class width.
Example 4.6: Suppose 20 wind measurements were taken in one area with the hope of
understanding the distribution of wind speeds.
The Data:
17.7
11
24.3
12.9
20.7
24.1
23.7
21
16.9
18.9
27.1
10.2
14.6
15.5
28.2
18.8
20.1
15.7
14.1
23.3
Solution 4.6:
Determine the class boundaries
Lowest Value = 10.2
Highest Value = 28.2
Use 10 for the starting value and use a class width of 5.
Create the frequency distribution
Classes
Frequency
[10,15)
5
[15,20)
6
[20,25)
7
[25,30)
2
Steilacoom Valley Wind Speeds
9
8
7
No. of obs.
6
Draw the histogram
5
4
3
2
1
0
10
15
20
25
30
X < Category Boundary
Wind Speed (MPH)
This graph suggests that wind speeds exceed 10 miles per hour, and are frequently
between 20 and 25 miles per hour. This would make a good location for windmills.
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STATISTICS FOR QUANTITATIVE DATA
For quantitative data, we are interested in two different statistics. One is a number to
represent the center of the data set and the other is a number to represent the variation in the data.
The most common measures of the center are the arithmetic mean and median. Up to this
point we have been using the word average, but now we will be more formal and call it the
mean.
The mean of a set of n observations of a quantitative variable is simply the sum of the
observation values divided by the number of observations, n.
Sample Mean: x
x
n
x
N
The symbol ∑ is an upper case Sigma and means summation – to add up all the data values.
Population Mean:
Example 4.7: Find the mean of the following three sets and show on a number line.
Set A 1,2,3; Set B 1,2,6; Set C 1,2,12
Solution 4.7:
6
2
3
9
x 3
Set B 1,2,6
3
15
5
Set C 1,2,12 x
3
Set A 1,2,3
0
A
B
C
1
x
A
B
C
2
A
B
3
4
5
6
7
8
9
10
11
C
12
The means are shown with a circle. Notice that the mean is not always a good
representation of the data set.
Median – The median of a set of n observations, ordered from smallest to largest, is a value such
that at least half of the observations are less than or equal to that value and at least half the
observations are greater than or equal to that value.
Math In A Sustainable Society 2.2
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n 1
to find which value is in the middle
2
where n is the number of data values once they have been put in order from lowest to highest.
The median is the middle value of ordered data. Use
Example 4.8: Find the median of 10, 8, 11, 3 and 12.
Solution 4.8: Put the numbers in order: 3,8,10,11,12. Since there are 5 numbers then
5 1
3 . This means the median is the third number, which is 10.
2
n 1
=
2
Example 4.9: Then find the median value of 5, 25, 8, 10, 20, and 16.
Solution 4.9: Put the numbers in order: 5,8,10,16,20,25. Since there are 6 numbers then
n 1
=
2
6 1
3.5 . This means the median is halfway between the third number, which is 10 and the
2
fourth number which is 16. Thus, the median is 13.
STANDARD DEVIATION
In addition to finding the center of a data set, we also need some idea of the spread of the
data. This is determined by calculating the standard deviation. Standard deviation is
approximately the average distance between each point and the mean. Consider the following
two sets of data:
Set 1: 4,5,6,7,8
Set 2: 1,2,6,10,11
Make a frequency plot of both. What is the mean and median?
Do they look the same? Notice that set 2 values are spread out more than set 1 values.
We would expect that the average distance each value in set 2 is from the mean is greater than
the average distance each value in set 1 is from the mean. We calculate this using the formula
for standard deviation. As with the other statistics, there is a difference in notation between the
standard deviation of the population and the standard deviation of the sample. The two formulas
are
Population standard deviation: σ σ
2
Sample standard deviation: s =
s2
x
2
N
x x
n 1
2
Because the population standard deviation (σ – lower case sigma) requires knowledge of
μ, which would require a census, we will focus on the sample standard deviation (s).
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When calculating standard deviation by hand, it is convenient to use a table. The first
column in the table is for the data, the second column shows the difference between the mean
and the data, the third column shows the square of the difference. The sum of the third column
becomes the numerator for the formula.
x
xx
x x 2
4
5
6
7
8
4-6 = -2
5-6 = -1
6-6 = 0
7-6 = 1
8-6 = 2
4
1
0
1
4
Total: 10
s
x x
n 1
2
10
5 1
s = 1.58
s
Determine the standard deviation for set 2.
x
xx
x x 2
1
2
6
10
11
Total:
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In-Class Activity 4.2: QAW Histogram
Name___________________________ Points ______/ 6 Attendance ___/ 2
Total ______/8
During the QAW project at the beginning of the quarter, all students calculated the
weighted mean for their evaluation of critical issues that were presented graphically. Enter the
scores for the class in the table below.
Make a frequency distribution and histogram of the data. Use reader - friendly classes,
keeping in mind that the scoring system was meant to be consistent with the grading system used
in schools (4 point scale). After making the histogram in the space below, find the mean and
median.
Mean_______
Median ______
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This Page Is Available For Notes, Doodling, Ideas or Computations.
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THEORY
We now know how to find the mean and standard deviation of sample data, but to answer
any question that would be the reason for conducting research in the first place requires that we
use the knowledge of the sample to infer something about the mean of the population. Our
ability to do this will be based on the following concepts.
1. Mathematical models can be used to represent irregular or unknown distributions
2. The Central Limit Theorem allows for the use of the Normal distribution and leads to a
confidence interval formula for estimating the parameter.
MATHEMATICAL MODELS
If you were asked to find the area of the following shape, how would you do it?
One approach would be to cover it with equally spaced grid lines then count the number
of squares.
An alternate approach is to model the shape with a geometric shape of known properties.
In this case, since the shape looks somewhat like a circle, we might model it with a circle.
With this circle, we can find the radius and then use the formula A = πr2 to estimate the
area of the original drawing. We may not have a precise area for the original drawing, but it
should be a close estimate.
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Extending this concept to data, a smooth curve has been placed on the graph that was
presented in figure 4.1. The graph, with the curve is reproduced in Figure 4.4. The curve is
called the Normal Distribution but also goes by the name bell curve.
Figure 4.4
Distribution Of All Of The 729 Possible Sample Means When 3 Units Are Selected, With Replacement,
From The Population Of 15,20,20,25,25,25,30,30,35
180
160
140
No of obs
120
100
80
60
40
20
0
15.0000
18.3333
21.6667
25.0000
28.3333
31.6667
35.0000
16.6667
20.0000
23.3333
26.6667
30.0000
33.3333
Like the circle in the earlier example, this curve has some known properties. Among
these properties are
1. The area under the curve is 1.
2. The mean and median are both located in the middle of the distribution. Half the
curve is above the mean and half is below the mean.
3. The curve can be labeled with 3 standard deviations on either side of the mean.
4. Approximately 68% of the curve is located within 1 standard deviation of the mean.
5. Approximately 95% of the curve is located within 2 standard deviations of the mean.
6. Approximately 99.7% of the curve is located within 3 standard deviations of the mean.
7. The area of a portion of the curve corresponds to the probability of selecting a value
within that portion.
68%
95%
99.7%
-3
-2
-1
0
1
2
3
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SAMPLING DISTRIBUTION OF SAMPLE MEANS
When you learned to make a histogram of data, you were making a histogram of
individual values of the random variable X. If, on the other hand, you took samples of size n
from the population and found the sample means and made a histogram of the sample means, x ,
that would be a sampling distribution of sample means. While it is not logical to create sampling
distributions, it is necessary to visualize them to understand statistical inference.
Figure 4.5 shows the distribution of 1330 wind speeds, which are raw data. Notice that
the data are not normally distributed and consequently is not well modeled by the normal curve.
Figure 4.5
If samples of size 36 are drawn from this distribution, then the means of these samples
can form a sampling distribution of sample means. This distribution, consisting of 1000
different sample means drawn from the population of the original wind speed data, is
approximately normally distributed. It is shown in Figure 4.6.
Figure 4.6
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CENTRAL LIMIT THEOREM
The Central Limit Theorem is one of the most important of all the statistical theorems.
This theorem states that given any distribution with a mean μ and a standard deviation σ, the
distribution of sample means will be normally distributed with mean μ and standard deviation
, provided the sample size is sufficiently large. What this implies is that regardless of the
n
shape of the distribution of the data, the distribution of sample means will be normal. In general
this is the case if the sample size is greater than 30. If the sample size is less than or equal to 30,
the original data must be normally distributed.
Figures 4.5 and 4.6 illustrate the Central Limit Theorem. In 4.5, the data are not
normally distributed, but in 4.6 the sample means are normally distributed. By putting the
histogram from 4.6 onto the same x-axis scale as in 4.5, it is evident in Figure 4.7 that the
standard deviation of the sample means is much less than the standard deviation of the original
data. This is shown because the curve is narrower.
Figure 4.7
CONFIDENCE INTERVALS
Since the distribution of sample means is normally distributed with mean μ and standard
deviation
and 95% of a normal curve falls within two standard deviations, then we can
n
conclude that 95% of all possible sample means from a population will fall within 2 standard
deviations of the mean of the population. Conversely, the mean of the population should then be
within two standard deviations of 95% of all the possible sample means we could get. Therefore,
if we start with the sample mean, which would be determined from the sample data and add and
subtract 2 standard deviations, then we create an interval that has a good chance of containing
Math In A Sustainable Society 2.2
107
the parameter μ. Mathematically this is shown as x 2
. There is one slight problem with
n
this formula. We don’t know the value of σ. Therefore, we estimate its value with s, the sample
s
standard deviation. The formula we use for finding the 95% confidence interval is x 2
.
n
Example 4.10: To foster a sense of community in Steilacoom Valley and to reduce energy
demand, meals are eaten together in group dining areas. A dining area manager wants to know
the average number of people who eat their breakfast at that dining area so that they can prepare
enough food, without producing too much waste. A random sample of 36 days from the prior
year shows the number of people at breakfast.
213
183
282
138
321
153
214
163
147
216
227
125
260
152
253
239
190
293
171
218
252
173
199
124
257
187
171
181
115
117
196
219
155
212
127
267
The mean of this data is 197.5. The sample standard deviation is 53.2. The sample size is 36.
Solution 4.10:
The 95% confidence interval is x 2
s
197.5 2
n
53.2
36
197.5 17.7 or 179.8 < μ < 215.2
Conclusion: About 95% of the sample means we could get produce a confidence interval that
will contain the mean of the population. Based on our sample, we estimate that the mean
number of people who come for breakfast is between 179.8 and 215.2.
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This Page Is Available For Notes, Doodling, Ideas or Computations.
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In-Class Activity 4.3: Analyzing Quantitative Data
Name_______________________________Effort____/3Attendance ___/1 Total ___/4
Suppose that instead of the formal educational system used in much of the world, all
people were encouraged to be active learners, regardless of age. To achieve that goal, the
community’s library became a major resource. Everyone, regardless of age, was encouraged to
read whatever they wanted. There were no recommended books (other than by friends) and no
book was considered too advanced or too simple for anyone. There were no book reports or
exams. The librarians kept record of the number of books read per month by all the residents
that were old enough to read. A sample of this data is provided in the table below.
1
23
9
14
6
4
1.
2.
3.
4.
5.
0
10
3
11
5
4
26
5
8
5
10
6
3
10
20
1
18
17
7
25
17
19
5
9
Make a frequency distribution for the number of books read per month
Make a complete histogram for the number of books read per month
Find the mean
Find the median
The sample standard deviation is 7.4. Find the 95% confidence interval.
[0,5)
[5,10)
[10,15)
[15,20)
[20,25)
[25,30)
7
10
5
4
2
2
Mean: 10.03
Median: 8.5
The 95% confidence interval is x 2
s
n
7.4
10.03 2
28
10.03 2.80 or 7.23 < μ < 12.83
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110
This Page Is Available For Notes, Doodling, Ideas or Computations.
Math In A Sustainable Society 2.2
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GRAPHS AND STATISTICS FOR QUALITATIVE (CATEGORICAL) DATA
Winds need to be over 10 mph to cause the blades of a windmill to spin. If the windmill
is not spinning, electricity is not being produced. If windmills are the sole source of electrical
energy, then the amount of time the turbine doesn’t spin because the winds are too slow is an
issue of concern. Suppose the engineers want to locate the windmills in a place where the winds
are too slow only 10% of the time. They monitored winds in one location and found that out of a
sample of 200 wind measurements taken at 150 above the ground, 53 were less than 10 mph.
One graph that is used for qualitative data is a pie chart. To make a pie chart, determine
the proportion of each category then, if creating the graph by hand, divide the pie chart into
sections of approximately that size. To make a reasonable estimate, mentally divide the pie into
quarters (25%) then use that as the basis for your lines. The statistic, which is the sample
x
proportion is given by pˆ .
n
Category
x
n
Slow winds
<10mph
Fast winds
≥10mph
53
200
147
200
Sample
Proportion
53
.265
200
147
.735
200
This is shown in the graph in Figure 4.8.
Figure 4.8
Steilacoom Valley Wind Speeds at 150 Feet above Ground Level
10 mph or less, 53, 27%
Greater Than 10 mph, 147, 74%
From this graph, it does not appear that the wind measurements were taken in a good
location for a windmill because the winds were too slow 27% of the time, which far exceeds the
engineer’s desire. However, we must realize that this is sample data, it is not the parameter.
Therefore we need to use these results to estimate the parameter which is the proportion of all
wind speeds that are too slow.
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112
THEORY
The theory that allows for the estimation of a population proportion is different than the
theory that allows for the estimation of a population mean, although both are based on the
concept of sampling distributions. For proportions we use the distribution of sample proportions.
The sample proportions are the proportion of successes for a particular sample size. To create
these distributions we need to remember the three probability rules discussed earlier.
P( x)
number of favorable outcomes
, P(A and B) = P(A)P(B), P( A ) = 1-P(A).
number of possible outcomes
Instead of presenting this in terms of coin flips that yield heads or tails, the concept will
be presented more generally using the terms success and failure. These terms are not used in the
traditional sense of success being good and failure being bad. Usually success is based upon the
research question. If the question was “what proportion of the native salmon has sea lice as a
result of a nearby salmon farm” then success would be considered the salmon with sea lice even
though that is a bad thing.
Our ability to understand the characteristics of a sampling distribution require us to first
use a distribution for which the proportion of successes is known. From this we will be able to
see the possible sample proportions that can be obtained. We will then be able to use a sample
proportion as a way of estimating an unknown population proportion. The process will be
explained first using counts of successes, but will conclude by using proportion of successes.
Suppose 60% of a population can be called a success. Then 40% of that population
would be a failure. If a sample of size 2 was made from this population, then the sample space
could be shown as {SS,SF,FS,FF}. The probability of each would be calculated using the
formula P(A and B) = P(A)P(B).
P(SS) = P(S)P(S) = (0.6)(0.6) = 0.36
P(SF) = P(S)P(F) = (0.6)(0.4) = 0.24
P(FS) = P(F)P(S) = (0.4)(0.6) = 0.24
P(FF) = P(F)P(F) = (0.4)(0.4) = 0.16
The probability of 0 successes is 0.16, the probability of 1 success is 0.48 (0.24 + 0.24),
the probability of 2 successes is 0.36. This is a small example to illustrate the point.
x
We can think of the number of successes in terms of proportions, where p . If there
N
0
are no successes in 2 selections, then p 0 ; the probability that p=0 is 0.16. If there is one
2
1
success in 2 selections, then p 0.5 ; the probability that p=0.5 is 0.48. If there are two
2
2
successes in 2 selections, then p 1 ; the probability that p = 1 is 0.36.
2
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113
A larger example was shown earlier in the chapter and is reproduced below (Figure 4.9).
It shows all possible results of flipping a coin 10 times. In this example, tails are considered a
success. To interpret the graph it is helpful to know that there is one occurrences for having 0
tails ( p 0 ) which can be shown as P(HHHHHHHHHH). The second bar represents 1 tail
( p 0.1 ) which can be shown in 10 different ways, such as P(THHHHHHHHH) or
P(HTHHHHHHHH) etc. The remainder of the bars represents other combinations of heads and
tails. Ultimately, there are 1024 different possible ways that the heads and tails can be ordered.
Figure 4.9
D istribution Of The Proportion Of Tails If A C oin Is F lipped 10 Tim es
C oin F lip = 1024*0.1*norm al(x, 0.5, 0.1582)
280
260
240
220
200
No of obs
180
160
140
120
100
80
60
40
20
0
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Proportion of Tails
CONFIDENCE INTERVAL FOR PROPORTIONS
Obviously, it could be very tedious to figure out all these possibilities. It would be nearly
impossible to do so for a large sample. Fortunately, we notice that the distribution of sample
proportions that are formed can be approximated with a normal distribution. The mean of the
normal distribution is equal to the population proportion p. The standard deviation of the normal
p(1 p)
distribution is
. Using the same logic as we did with creating a confidence interval
n
for the mean, we conclude that 95% of all sample proportions are within 2 standard deviations of
the population proportion. Conversely, the population proportion is within 2 standard deviations
p(1 p)
of 95% of all the sample proportions. This can be shown with the formula pˆ 2
.
n
Since we don’t know the value of p, we estimate it with the sample proportion p̂ . This results
in the formula we will use for the confidence interval for a population proportion,
pˆ (1 pˆ )
.
pˆ 2
n
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114
Example 4.11: Find the 95% confidence interval for the proportion of wind speeds that were
too slow if 53 out of 200 wind speeds were too slow.
Solution 4.11:
x
53
pˆ
= 0.265
n
200
pˆ (1 pˆ )
n
0.265(1 0.265)
0.265 2
200
pˆ 2
0.265 0.062 or 0.203 < p < 0.327. Thus we estimate the proportion of all wind speeds that are
too slow is between 20.3% and 32.7%. Since a location should have less than 10% of the wind
speeds that are too slow, this location is not desirable.
Math In A Sustainable Society 2.2
115
In-Class Activity 4.4: Analyzing Qualitative Data
Name _________________________Effort____/3 Attendance ___/1 Total ___/4
Gallup regularly asks the question: With which one of these statements about the
environment and the economy do you most agree – “protection of the environment should be
given priority, even at the risk of curbing economic growth (or) economic growth should be
given priority, even if the environment suffers to some extent? (Gallup 2010)
From 1985 through 2008, more people favored the environment, however the survey in
2009 showed that more favored the economy. While the actual numbers are not available from
Gallup, based on the statistics they provide we can use relatively realistic numbers. Suppose that
1200 people were surveyed and had an opinion and 542 of these favored the environment.
1. Make a pie chart.
2. What is the sample proportion of those who favor the environment?
3. What is the 95% confidence interval for the proportion of people in the country who favor the
environment?
Gallup Poll - Which is Favored Environment or Economy
Environment, 542, 45%
Economy, 658, 55%
pˆ
x
n
542
1200
= 0.452
pˆ (1 pˆ )
n
0.452(1 0.452)
0.452 2
1200
0.452 ±0.029
0.423 < p < 0.481
pˆ 2
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Chapter 4 Homework
Name_________________________________________ Points ___/___
1. As the manager of a store that hopes to sell solar panels, you have the responsibility of
comparing two different brands of panels, Apollo brand and Maui brand. (Apollo is the Greek
sun god and Maui is the Polynesian sun god). Both brands are rated as 1 kilowatt systems. Your
goal is to determine which of the two produces the most energy after at least one year of use. To
find out, you will contact people who have the system on their home. The data you will collect
is the maximum daily energy output (watts). You would like an estimate of the mean energy
production by each brand.
1a. Is this an observational study or an experiment? Observational study
1b. Is the data quantitative or qualitative? quantitative
1c. How many factors are there?
1d. How many levels are there?
1
2
1e. Complete the design layout table.
Factor:
Level 1
Level 2
Level 3
Response Variable
Parameter of interest
brands
Apollo
Maui
Energy output
mean
The data for the Apollo brand is 956, 890, 1000, 988 and 966.
1f. Find the mean 960
1g. Find the sample standard deviation 42.83
1h. Find the median 966
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2. One way in which people will survive peak oil is by developing transition towns. In these
towns, people will work together as a community to grow and preserve food and to share tools
and other resources. Transition towns involve a different psychology than our current towns in
which each person is focused primarily on the needs of their family, rather than of the
community. As long as gas is relatively inexpensive, most people in a town won’t want to
consider changes, but if the price goes up and the economy crashes, more people will look for
alternative means of surviving.
Suppose a community decided that any edible food that is grown in front of a house could
be picked and used by any community member while any food in the back of the house would be
reserved for the home owner. This would be a very strange concept in the US because private
property and fear of law suits would make people scared. However, assume this community
solved the law suits problem and 44 home owners started planting vegetables in their front yard.
Of these, 22 were randomly selected to have a sign in the front yard inviting people to help
themselves when the vegetables are ready for harvest. The remaining 22 did not have the sign.
The objective is to find if the signs are necessary to encourage people to pick the vegetables.
Each home owner kept track of the number of visitors they had during the summer. The goal
was to determine if the signs made a difference by comparing the average number of visitors at
the homes with signs to the average number of visitors at the homes without signs.
2a. If the 44 home owners were assigned a number between 01 and 44, what would be the first 5
randomly selected numbers starting in row three of the table of random digits in Figure 4.3.
26, 24, 23
2b. Is this an observational study or an experiment? Experiment
2c. Complete the design layout table using the underlined words.
Factor:
Signs
Level 1
With signs
Level 2
Without signs
Level 3
Response Variable
Number of visitors
Parameter of interest
average
2d. If the data consists of the number of people who pick vegetables at each house, is the data
qualitative or quantitative? quantitative
In one neighborhood there were 8 homes with gardens. The number of visitors who picked
vegetables at these homes was 3, 12, 0, 5, 0, 8, 15, 13.
2e. What is the sample mean number of visitors? 7
2f. What is the median number of visitors? 6.5
2g. What is the sample standard deviation for the number of visitors? 5.9
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3. For those who have adapted to the high stress, achievement oriented, modern day world, life
in Steilacoom Valley might seem rather slow. There would probably be more leisure time than
we are used to and TV would not be available for filling the time. A psychology researcher
wants to find how the residents are coping with a less stressed life. Assume there are 800 adult
residents and the researcher wants to randomly select 150 residents from this group. Each
resident is given a number between 001 and 800. The following two questions were on the
survey.
1. Do you want to return to the pace of life you had before moving to Steilacoom Valley?
2. How much time did you spend yesterday in conversation with your spouse/partner?
3a. Use Table 4.3, row 8 to randomly select the numbers of the first 3 people who will be asked
to take the survey.
3b. Is this an observational study or an experiment? Observational Study
3c. Is the data in question 1 qualitative or quantitative? qualitative
3d. What is the symbol x , , pˆ , p for the parameter of interest in question 1?
3e. What is the symbol for the statistic that can be found for question 1?
p
p̂
3f. Is the data in question 2 qualitative or quantitative? quantitative
3g What is the symbol for the parameter of interest in question 2?
μ
3h. What is the symbol for the statistic that can be found for question 2?
x
3i. The data given in response to question 1 was 25 out of 150 people answered yes. What
proportion of the people in the sample want to return to the pace of life they had before moving
to SV?
0.167
The data given in response to question 2 was: 10, 130, 70, 100, and 80.
3j. What is the mean amount of time couples spent in conversation? 78 minutes
3k. What is the median amount of time couples spent in conversation? 80 minutes
3l. What is the sample standard deviation of the amount of time couples spent in conversation?
44.38 minutes
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4. In Steilacoom Valley, careful records are kept of the agricultural yield. The original
6000C
expectation was that 6000 calories could be produced per day per acre,
. With about
day acre
1000 acres of farmland, an estimate of the total yield can be obtained by sampling 15 acres. The
results of this sampling are shown in the table below. The residents of SV have a particular
interest in this because it affects their year long food supply.
5285
5709
6551
6680
6763
5244
5940
5666
5876
5382
4525
5711
6157
6677
5563
4a. Make a frequency distribution and histogram for this sample data.
Caloric Yield of Steilacoom Valley Farmland
7
1
3
6
1
4
6
5
No of obs
[4500,5000)
[5000,5500)
[5500,6000)
[6000,6500)
[6500,7000)
4
3
2
1
0
4000
4500
5000
5500
6000
6500
7000
Caloric Yield (Calories/Day*Acre)
4b. Find the sample mean. 5848.6
4c. If the sample standard deviation is 633.4, what is the 95% confidence interval for the mean
yield?
x2
s
n
5848 .6 2
633 .4
5848.6 ± 327.1 or 5521.5 < μ < 6175.7
15
4d. As a resident of SV, what do you think are the implications of this result?
Since the confidence interval includes values that are less than 6000, upon which carrying
capacity was based, it is possible that the average yield will not be enough, leading to insufficient
food for the residents.
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5. The Steilacoom River has an abundant salmon run each year. While salmon are a good
source of protein and omega oil, fishing sustainably is important for the resource to be available
in future years. Consequently, the community strongly supports the concept of not keeping the
largest salmon. Since fishing will only be done for food, not for trophies to put on the wall,
leaving the biggest salmon to spawn will mean that the genes of large salmon will be passed on
to future generations. If these fish were removed, then over time, the salmon would become
smaller. Out of 136 salmon that were caught this year, 28 were returned because they were too
big.
5a. Make a pie chart for this data
Salmon Size
Too Big, 28,
21%
Good Size,
108, 79%
5b. What is the sample proportion? 0.206
5c. What is the 95% confidence interval for the proportion of all the salmon in the run that are
too big?
p(1 p)
0.206 (1 0.206 )
0.206 2
.206 ± 0.069 or .137 ≤ p ≤ .275
pˆ 2
136
n
6. The Steilacoom Valley Bakery attempts to produce just enough bread so that there is no
waste. Unfortunately, that means that some days there won’t be enough bread for everyone. Of
286 randomly selected days, there were 42 days in which there was surplus bread at the end of
the day.
6a. Make a pie chart for this data
Days With Surplus Bread
Surplus, 42,
15%
No Surplus,
244, 85%
6b. What is the sample proportion? 0.147
6c. What is the 95% confidence interval for the proportion of all days there are surplus bread?
p(1 p)
0.147 (1 0.147 )
0.147 2
.147 ± 0.042 or .105 ≤ p ≤ .189
pˆ 2
286
n
Math In A Sustainable Society 2.2
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7. In Chapter 3, it was estimated that each home in SV will consume an average of 12.3 kWh
per day. A random sample of homes resulted in the following set of data.
19.0
21.1
16.0
18.1
13.0
17.3
19.0
16.5
12.8
20.3
13.5
16.9
17.1
11.5
17.8
19.0
18.7
17.4
19.5
15.1
7a. Make a histogram for this sample data.
Daily Household Energy Consumption
8
1
3
1
7
6
2
7
6
5
No. of obs.
[10,12)
[12,14)
[14,16)
[16,18)
[18,20)
[20,22)
4
3
2
1
0
10
12
14
16
18
20
22
Energy Consumption (kWh)
7b. Find the sample mean. 16.98 ≈ 17.0
7c. The sample standard deviation is 2.6; what is the 95% confidence interval for the mean
amount of energy?
x2
s
n
17 .0 2
2.60
17.0 ± 1.2 or 15.8 < μ < 18.2
20
7d. As a resident of SV, what do you think are the implications of this result with regards to the
number of windmills we planned for?
The number of windmills was based on the assumption that houses used an average of 12.3 kWh
per day. Because even the low end of the confidence interval is greater than 12.3, it is very
likely that the actual average energy consumption is greater than 12.3. This would mean the
demand for electricity will exceed the supply, so more windmills may be needed.
Math In A Sustainable Society 2.2
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Chapter 5 System Dynamics Modeling
The world in which we live is very complex. Most news items look at one issue in
isolation from all other issues. For example, health care is not discussed in the same report as
wealth distribution. The growth of the gross national product is not discussed in the same report
as resource depletion. Likewise, worldwide hunger is not discussed with global climate change.
This book started with the Quantitative Assessment of the World project as a way to
begin the process of thinking of multiple issues at the same time. During the Consequences
project you had the opportunity to think about systems and create your own system designs.
This chapter will introduce you to system dynamics modeling, a formal method for creating
models to improve understanding of large natural and manmade systems such as global climate
change and resource depletion.
Because systems can be very complex, we use models to simplify and understand them.
System Dynamics makes use of stock and flow models. A demonstration of a model will be
made that shows the growth of a person’s checking account.
Stocks represent accumulation or storage in a system. The stocks that are to be
monitored are shown with a rectangle. For this example, the stock will represent the amount of
Checking Account
money in the checking account. It is shown as:
Stocks are also shown with clouds, however we have no particular information about the
clouds and so they primarily serve as a reminder that the system we are analyzing connects to a
bigger system.
Flows are what cause a change to a stock. For our example, the flows are deposits and
withdrawals (writing checks). Flows are shown as arrows with a labeled circle that goes into, or
out of, a stock.
Checking Account
Deposits
Withdrawals
There are many potential things that affect the flows. These are identified with
converters. Since a model is meant to be a simplification of a system, care must be taken to not
make it too complicated. One converter is used for the deposits. This converter represents a pay
check. Four converters are used for withdrawals. These are housing, food, utilities and other.
Math In A Sustainable Society 2.2
124
Checking Account
Deposits
Pay Check
Withdrawals
Housing
Food
Utilities
Other
By identifying the components of a system, creating formulas and inserting numbers, it is
possible to understand the changes a stock experiences over time.
One of the things that these models can show is feedback. Feedback can be positive
(reinforcing) or negative (balancing). Positive feedback means that a change in the stock causes
a change in the flow that resulted in more change in the stock. An example is an account that
increases in size which results in an increase in interest which leads to another increase in size.
We discussed these accounts in Chapter 1 while learning about compound interest. A negative
feedback loop creates a change in the opposite direction. For example, if the amount of money
you have increases enough, you will be tempted to spend more of it, thereby resulting in a
decrease to the money you have.
A simple system to understand the population of an organism (for example, rabbits) can
be shown as:
Population
of an organism
Births
Deaths
Death Rate
Birth Rate
This system contains both a positive and a negative feedback loop. The birth/population
loop is a positive feedback loop because an increase in births causes an increase in the organism.
Having more organisms will result in more births as well. Since the changes are in the same
direction, each relationship (Births to population and population to Births) is given a positive
sign. The entire loop is given a positive sign because there are no negative signs for the
individual relationships.
+
Births
+
+
Population
-
-
+
Deaths
The death/population loop is a negative feedback loop because an increase in deaths
causes a decrease in the population while an increase in population causes an increase in deaths.
In the death to population relationship, the changes are in the opposite direction, thus we give
that relationship a negative sign. The population to death relationship is in the same direction so
it receives a positive sign. Since there are an odd number of negative signs, the entire loop is
given a negative sign to indicate a negative feedback loop.
Math In A Sustainable Society 2.2
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In-Class Activity 5.1: Stock and Flow Model
Name_____________________________ Points ______/3 Attendance ____/1 Total _____/4
Create a stock and flow model for the community food supply using the following elements.
Stock:
Flows:
Converters:
Available Food
Food Produced
Food Consumed
Wild Game and Fish
Fruits, Nuts, Vegetables and Grains
Raised Meat (Chicken, etc)
Eating
Spoilage
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COMPUTER MODELING
The stocks, flows and converters are connected through mathematical equations. These
equations allow the change in a stock to be shown as a function of time. From this, a graph can
be produced to show the stock’s behavior.
The first example about the checking account balance is shown with Stella software. We
can start with some assumptions and see how the checking account balance changes over time.
In this case, we will simulate pay checks of $1800, $1900, $2000 and $2100 per month (take
home pay). We will assume the checking account has $1000 at the beginning.
Assumptions: Housing: $1200 per month
Food: $300 per month
Utilities: $150 per month
Other: $250 per month
Checking Account
Deposits
Withdrawals
Pay Check
Housing
Utilities
Food
Other
1 = $1800, 2 = $1900, 3 = $2000, 4 = $2100
Checking Account: 1 - 2 - 3 - 4 1:
9000
4
4
1:
4500
3
4
3
3
4
3
1
1:
0
0.00
Page 2
2
2
1
9.00
2
1
18.00
Months
2
1
27.00
36.00
7:01 PM Mon, Dec 06, 2010
Notice that the $1800 paycheck leads to depletion of the checking account because more
money is being spent each month than is being earned. The $1900 pay checks produces is a
horizontal line because deposits and withdrawals are equal. The other two lines show a growth
in the checking account balance because more money is deposited each month than is
withdrawn.
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Because Stella software may not be available on the school’s computers, we will use
Microsoft Excel to model a system. The first example will be a finance model that is based on a
one time deposit of $1000. The model is shown below. It should remind you of topics that were
discussed in Chapter 1. We will design the spreadsheet to calculate the value of the stock
(account balance) for the first 30 years. These values will be computed for three interest rates,
3%, 5% and 7%, compounded annually. Ultimately, we will graph the account balances for all
three interest rates on the same graph so we can compare the results.
Account
Balance
Interest
Added
Interest Rate
Make a causal loop diagram for the stock and flow model.
Math In A Sustainable Society 2.2
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Sample Spreadsheet Design for Comparing the Account Balance of an Investment
A
1
2
3
4
5
B
Constant
Interest Rate
Compounding
frequency
0.03
C
D
Flows
E
Interest Added
Years
G
Stock
Account
Balance
=G3*((1+$B$2/$B$3)^$B$3-1)
=G4*((1+$B$2/$B$3)^$B$3-1)
0
1
2
1000
=G3+D4
=G4+D5
12
F
Time
H
I
J
K
L
M
Comparison of balances with
different interest rates
Balance Balance Balance
Years (r=0.03) (r=0.05) (r=0.07)
After typing in the formulas, highlight D4:G5 then autofill until you reach year 30, which will occur in row 33. Copy and paste the
year values in column J then copy and paste special – values the account balances in column K. Change the interest rate to 0.05 then
copy and paste special – values the account balances in column L. Repeat with a 7% interest rate and paste into column M. Make an
xy scatter graph of columns J-M. Label completely.
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Steilacoom Valley Population
Modeling the population in Steilacoom Valley is similar to, yet somewhat different than
the financial model. The financial model only includes the addition of interest. For the
population model, it is necessary to consider births and immigration which add people to the
Valley and deaths and emigration which remove people from the valley. For this modeling, we
will assume the immigrants will only be accepted as replacement for emigrants, consequently the
two values will be equal. We will also assume that death rates remain constant.
The primary complication with this model is birth rates. Birth rates tend to decrease as
the carrying capacity is reached. Remember that we found the carrying capacity of Steilacoom
Valley to be 2494? For our model, we will round this to 2500. We need a way to model the
changing birth rates. This will be done by multiplying the birth rate by a value that varies from 1
to 0 and decreases as the population increases. The usual approach would be to use a linear
model for the multiplier. We will compare that to an exponential decay model. To do so, we
will generate the equation of the line that connects the points (0,1) to the point (2500,0.001). As
ordered pairs, the x coordinate represents the population and the y coordinate represents the
multiplier.
Notice that both these points have some flaws that we will ignore, but they will still
provide a good estimate. Assume the birth rate of 5%. The first point says that if there are no
people in Steilacoom Valley, the birth rate will still be 5%. However, if there are no people in
Steilacoom Valley, there obviously cannot be any birthrate. The other point (2500,0.001) would
ideally be written as (2500,0) because the birth rate would decline to zero when the carrying
capacity has been reached. However, since an exponential function cannot have a value of 0, it
is necessary to modify that value to 0.001. These issues do not alter the fact that these models
can still help us understand the system.
Figure 5.1
Multiplier = 1-0.0004*x
Multiplier = 1*exp(-0.0028*x)
1.2
1.0
0.8
Multiplier
Linear Fit
0.6
0.4
0.2
Exponental Fit
0.0
-0.2
-200
0
200
400
600
800
1000 1200 1400 1600 1800 2000 2200 2400 2600
Population
The equation for the linear model is Multiplier = 1 – 0.0004x. The equation for the
exponential model is Multiplier = 1e-0.0028x.
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Sample Spreadsheet Design for Comparing the Effect of Birth Rate Multipliers on Steilacoom Valley Population
1
2
3
4
5
A
Constant
Birth Rate
Death Rate
Immigrate Rate
Emigration Rate
B
C
E
Birth Rate Multiplier
D
Flows
People Added
0.05
0.01
0.01
0.01
=1-0.0004*(G3)
=1-0.0004*(G4)
=(C4*$B$2+$B$4)*G3
=(C5*$B$2+$B$4)*G4
=($B$3+$B$5)*G3
=($B$3+$B$5)*G4
People Leaving
F
Time
Years
0
1
2
G
Stock
Population
1000
=G3+D4-E4
=G4+D5-E5
After entering the formulas in the table, highlight C4:G5 then autofill until you have 200 years of computations. Copy the
Time and Population values from columns F and G and paste them in another column using paste special – values. Then change the
formula for the birth rate multiplier to the exponential model typing the formula: =exp(-0.0028*G3) into cell C4 and then autofill.
Graph both the linear model and the exponential model on the same graph, with years as the x axis.
Does the linear model result in a graph that most resembles exponential growth, logistic growth, neither of these?
Does the exponential model result in a graph that most resembles exponential growth, logistic growth, neither of these?
Using the linear model, one might expect the population to reach the carrying capacity, but it does not. The system reached a point of
dynamic equilibrium at about 80% of the carrying capacity because births and deaths were equal.
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Appendix
Math In A Sustainable Society 2.2 - Instructors Manual
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