Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Download workshop files at http://www.montanalearning.org/web/MMMGTP.html
Modeling Process
The multi-step process of mathematical modeling will be employed, see Table 1 and
figures 1 and 2.
Step 1. Understand the problem or the question asked.
Step 2. Make simplifying assumptions.
Step 3. Define all variables.
Step 4. Construct a model.
Step 5. Solve and interpret the model.
Step 6. Verify the model.
Step 7. Identify the strengths and weaknesses of our model.
Step 8. Implement and maintain the model for future use.
Table 1 Steps in the Mathematical Modeling Process
Figure 1. Modeling process to be used.
We will take the real world system and apply many simplifying assumptions so that we can
employ mathematical models to measure the effects. Figure 2 illustrates this.
Real World Systems
M athematical World
Models
Observed Behavior
Mathematical Operations
Mathematical Conclusions
Figure 2. Mathematical Models and the Real World
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
HiMCM Problem A 2003
What is it worth?
In 1945, Noah Sentz died in a car accident and his estate was handled by the local courts.
The state law stated that 1/3 of all assets and property go to the wife and 2/3 of all assets
go to the children. There were four children. Over the next four years, three of the four
children sold their shares of the assets back to the mother for a sum of $1300 each. The
original total assets were mainly 75.43 acres of land. This week, the fourth child has sued
the estate for his rightful inheritance from the original probate ruling. The judge has ruled
in favor of the fourth son and has determined that he is rightfully due monetary
compensation. The judge has picked your group as the jury to determine the amount of
compensation.
Use the principles of mathematical modeling to build a model that enables you to
determine the compensation. Additionally, prepare a short one-page summary letter to the
court that explains your results. Assume the date is November 10, 2003.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Modeling with Discrete Dynamical Systems
Model a prescribed drug dosage.
One time drug: Novocain (Procainamide Hydrochloride), injected as an
anesthetic for minor surgical and dental procedures, is eliminated from the body
primarily by the kidneys. Loosely speaking, during any 1-hour period, the kidneys
take a fixed percentage of the blood and remove medicine from the blood. Let's
assume the kidneys purify 1/5 of the blood every one-hour period.
Let: u(n) = the amount of the prescribed drug in our system after n (one-hour)
periods.
u(n+1)= u(n) - .20 u(n) = .80 u(n), n = 0, 1, 2, 3, ….
Let's assume we are given 500 mg of the drug in period 0, so u(0)=500.
Let's iterate and graph the DDS to see what happens over a long period of time.
In this hand out we demonstrate how a TI-83 Plus can be used but you could
also use an EXCEL spreadsheet.
This model is a discrete dynamical system, DDS. Some background is required.
Using a DDS involved modeling with the paradigm:
Future = Present + Change
A DDS is a discrete function that can be used to model many situations, such as
Mortgage of a home
Car financing
Investment or financial alternatives
Prescribed drug dosages
Population dynamics
Predator-Prey
Competitor Hunter
Genetics
The above is known as a Discrete Dynamical Systems.
Calculus is the study of change, so
change = future value - present value
The above is known as a Difference Equation.
The following are Key Definitions/Concepts in modeling with discrete dynamical
systems:
A sequence is a function whose domain is the set of all nonnegative integers
and whose range is a subset of the real numbers.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
A dynamical system is a relationship among terms in a sequence.
A numerical solution is a table of values satisfying the dynamical system.
A recurrence relation is an equation of the form:
a(n+1)= f(a(n), a(n-1),...) where f is a discrete function.
A discrete dynamical system or difference equation is a sequence defined by a
recurrence relation.
Iteration is the process of obtaining a number in a sequence from previous
numbers.
Back to our example.
DDS on the TI-83 PLUS or TI-84
We can build the solution on the TI-83 Plus or TI-84.
TI-83 Calculators and DDS
1. Go to MODE, func, SEQ. SEQUENTIAL (2nd quit)
2. Go to y=
3. Put in our DDS:
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
4. Set the window for n [0,24], x [0,24], y [0,505]
5. Press Graph
We clearly see that this drug decays over time. There is less and less of the drug
in our system after each hour. Remember that tingling feeling as the feeling
begins to return to your face. Hopefully, we received this drug (via injection) only
one time.
More Than One Time Drug Dosage
Now, let’s consider a drug taken more often than once. CIPRO is a drug for
combating many infections, including anthrax. Let's assume that during a onehour period that our kidneys purify 1/4 of this drug from the blood. Let's assume
that the dosage is 16 mg each time period. Let’s see what happens between
each dosage.
Write the mathematical model that represents this system.
So our model is,
d(n+1)= .75 d (n), d(0)=16 mg, n=0,1,2,3,… (in one-hr periods)
Let's assume that to be effective, you must have at least 6.75 mg of the medicine
in your blood. How often should you take the medicine?
Now, let's remodel assuming that every 4-hour period we take 16 mg of the drug
and that we don't have any in our system when we begin, d(0)=0. Let’s assume
that in a 4-hour period the kidney’s only purify 60% of the drug.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
The model is: d(n+1)=d(n)-.6*d(n) + 16 or
d(n+1) = .40 * d(n) + 16 for n=0,1,2,3,…( n now represents 4-hour periods).
What happens now? Look at a long period of use for this drug. Describe what
you see from the graphical output.
This shows a stable equilibrium value. The definition of an equilibrium value is
when d(n+1)=d(n), INPUT=OUTPUT, and there is no change. This equilibrium
value is _____ mg of this drug. We will talk about the “stable” part later.
Extension: Is this what really happens? Model this in 1- hour periods starting
with an initial 16 mg dosage. Explain the graph.
CAR FINANCE EXAMPLE.
You want to buy a $20,000 new car and you can afford a monthly payment of
only $400 per month. As you shop around you find a dealer that will give you
$1,500 cash back if used as a down payment and 6.9% per year financing
compounded monthly. Can you get your new car?
The Model:
Let a(n) = the amount that you owe the financing company after n months
a(n+1)=a(n)+(.069/12) a(n) - 400
a(n+1) = (1 + .069/12) a(n) - 400
How can we solve this DDS? Well, one method is by numerical iteration. We
know that
a(0)=$18,500.
Go ahead and work through this problem.
Additional Problems:
1. Iterate to n=20 for each of the following DDS. Calculate the equilibrium value
for each DDS. For each obtain a graph and see if there appears to be a stable
equilibrium value.
a) a(n+1) = .2 a(n) + 6 ; a(0) = 2
b) a(n+1) = 1.2 a(n) + 3; a(0) = 0
c) a(n+1) = 1.2 a(n);
a(0) = 0
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
d) a(n+1) = - [a2(n)] + 6; try a(0) = 0, a(0) = 2, a(0)=-3
2. A sewage problem yielded the model:
s(n+1) = .88 s(n) n=0,1,2,... hours
s(0)=100 lbs
Answer the following using TI-83 or TI-84
a) What percentage is left after 1 day? How many hours did you assume per
day?
b) How long does it take to lower the amount of sewage by half?
c) How long until the amount of sewage is down 10% of its original level?
3. Consider a drug dosage problem for combating "anthrax". The doctor
prescribes 500 mg of Cipro as a daily dosage. We assume that the kidney's
diluted the drug by 45% every day. Model as a DDS and determine the long-term
behavior of the system.
Higher Order Discrete Dynamical Systems
Consider the following DDS,
A(n+2)= 0 .5*A(n+1)-0.25*A(n), n=0,1,2,3,
A(1)=6, A(0)=12
You must have 2 initial conditions for 2nd order equations.
WILL THE SPOTTED OWL SURVIVE?
Statement of the Problem:
There is a general fear that perhaps one quarter of the earth's current plant and animal
species will be extinct by the end of the century, primarily as a result of human activity.
At the center of a current controversy on the Pacific Coast of the United States is the
northern subspecies of the spotted owl.
The National Forest Management Act of 1976 legislates that timber harvesting on the
National Forest lands must be managed so that viable populations of native vertebrate
species are maintained over a wide domain. Home range territory for a nesting pair of
spotted owls consists of a huge tract of land, ranging from 2000 to 4200 acres of forest.
The Forest Service management plan calls for 500 to 600 spotted owl habitat areas, each
containing 1000 to 3000 acres of suitable habitat per nesting pair. This management plan
is at the center of a heated controversy between the timber industry and environmental
organizations.
The timber industry claims that management is too costly. Based on current values, the
land set aside for the spotted owls would yield about $2.2 billion in profit for the timber
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
industry. In general, the value of timber set aside to accommodate a single nesting pair is
about $8 million. To emphasize the magnitude of the economics, about 44% of Oregon's
economy and 28% of Washington's depend on national forest resources.
Environmentalists claim that continued logging of the forests will eventually drive the
owl to extinction. Many in the timber industry believe that the spotted owl is expendable
by virtue of its impact on jobs and local economies.
Our goal is to model the population of female spotted owls and predict what will happen
to the species.
You will need the following biological data on the spotted owl:
1.
The owl is a pulse breeder, with a short breeding season in late April to early May. Accordingly,
we adopt a discrete model to project the population from year to year.
2.
There are three distinct stages in the life of an owl:
a. juvenile (first year)
b. sub-adult (second year)
c. adult (third year and beyond)
Resulting Model:
Let J(n) = number of juvenile owl population after n years
S(n) = number of sub-adult owl population after n years
A(n) = number of adult owl population after n years
j = proportion of juveniles alive at n years who survive to become sub-adults at time n+1 years
s = proportion of sub-adults alive at n years who survive to become adults at time n+1 years
a = proportion of adults alive at n years who survive to become adults at time n+1 years
b = average number of juveniles produced by an adult female
The resulting difference equations are:
J(n) = b A(n)
S(n+1) = j J(n)
A(n+1) = s S(n) + a A(n)
1. Develop a second order difference equation that models the number of adult female
owls after n years.
2. Given that the adult spotted owl population was 1500 in 1989 and was 1400 in 1990,
write down initial conditions for this problem.
3. Several groups have studied this situation and have estimated the values for the
survival and birth parameters.
parameter
a
U.S. Forest Service
.97
Lande
.94
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
j
.34
.11
s
.97
.71
b
.24
.24
Using each set of data, what is the particular solution for the DDS? (May want to divide
the class into two groups - one for each set of data).
4. What is the equilibrium value for this Discrete Dynamical System? Are these
equilibrium values stable or unstable? Support your answer?
5. What is the long term behavior of the system for each of the data sets? Do these results
surprise you? What does this say about the survival of the spotted owl population?
Extended Problems:
6. An independent contractor studied the problem and came up with the following data: a
= .7, j = .75, s = .8, b = .5
According to this data, will the spotted owl population survive?
7. Suppose you accept the United States Forest Service data and are devising a
preservation program. Suppose your program focuses solely on improving the
survivorship of adult females. What is the minimum annual adult survivorship
(proportion of adults alive at time n who survive to become adults at time n+1) such that
the spotted owl population will not become extinct? How is the long term behavior of the
female spotted owl population affected if 50 female spotted owls are inserted into these
habitat areas each year? Write the corresponding difference equation which models this
scenario. Does this system have an equilibrium value? Is it stable or unstable?
8. Experiment with other parameters and determine their impact on the model and the
repercussions for a preservation program.
Nonlinear Discrete Dynamical Systems
Introduction
In this section we build nonlinear discrete dynamical systems to describe the
change in behavior of the quantities we study. To remind us let's define a
nonlinear DDS-- If the function of a(n) involves powers of a(n) (like a2(n)), or a
functional relationship (like a(n)/a(n-1)), we will say that the discrete dynamical
system is nonlinear. A sequence is a function whose domain is the set of nonnegative integers (n = 0, 1, 2...). We will restrict our model solution to the
numerical and graphical solutions. Analytical solutions may be studied in more
advanced mathematics courses.
Nonlinear Modeling Examples
Growth of a Yeast Culture
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
We often model population growth by assuming that the change in population is
directly proportional to the current size of the given population. This produces a
simple, first order DDS similar to those seen before. It might appear reasonable
at first examination, but the long-term behavior of growth without bound is
disturbing. Why would growth without bound of a yeast culture in a jar (or
controlled space) be alarming?
There are certain factors that affect population growth, such as resources
like food, oxygen, and space. These resources can support some maximum
population. As this number is approached, the change (or growth rate) should
decrease and the population should never exceed its resource supported
amount.
Problem Identification: Predict the growth of yeast in a controlled
environment as a function of the resources available and the current population.
Assumptions and Variables:
We assume that the population size is best described by the weight of the
biomass of the culture. We define y(n) as the population size of the yeast culture
after period n. There exists a maximum carry capacity, M, that is sustainable by
the resources available. The yeast culture is growing under the conditions
established.
Model:
y(n+1) = y(n) + k y(n) (M-y(n)) where
y(n) is the population size after n hours
n is the number of hours
k is the growth rate of the culture
M is the carrying capacity of our system
In our experiment, we find by data collection that the growth rate, k, is
approximately 0.00082 and the carrying capacity of the biomass is 665. This
model is
y(n+1) =y(n) + .00082 y(n) (665-y(n))
Again, this is nonlinear because of the y2(n) term. The solution iterated on the Ti83 Plus from an initial condition, biomass, of 9.6 is
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
The model shows stability in that the population (biomass) of the yeast culture
approaches 665 as n gets large. Thus, the population is eventually stable at
approximately 665 units. Try this on your calculator
Spread of a Contagious Disease
There are 1000 students in a college dormitory and some students have
been diagnosed with meningitis, a highly contagious disease. The health center
wants to build a model to determine how fast the disease will spread.
Problem Identification: Predict the number of students affected with
meningitis as a function of time.
Assumptions and Variables: Let m(n) be the number of students affected
with meningitis after n days. We assume all students are susceptible to the
disease. The possible interactions of infected and susceptible students are
proportional to their product (as an interaction term).
The model is,
m(n+1) - m(n) = k m(n) (1000-m(n)) or
m(n+1) = m(n) + k m(n) (1000-m(n))
It is found that two students returned from spring break with meningitis. The rate
of spreading per day is characterized by k=0.0025. It is assumed that a vaccine
can be in place and students vaccinated within 1-2 weeks.
The results clearly show that most students will be affected within 2 weeks. Since
only about 10% will be affected within one week, every effort must be made to
get the vaccination at the school and get the students vaccinated within one
week.
Exercises
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Consider the model a(n+1) = r a(n) (1-a(n)). Let a(0)=0.2. Determine the
numerical and graphical solution for the following values of r. Find the pattern in
the solution. Find Chaos.
1. r= 2
2. r=3
3. r=3.6
4. r=3.7
Projects
1. Consider the contagious disease as the Ebola virus. Look up on the
Internet and find out some information about how deadly this virus actually
is. Now consider an animal research laboratory in Restin, VA, a suburb of
Washington, DC with population 856,900 people. A monkey with the Ebola
virus has escaped its captivity and infected one employee (unknown at the
time) during its escape. This employee reports to University hospital later
with Ebola symptoms. The Infectious Disease Center (IDC) in Atlanta gets
a call and begins to model the spread of the disease. Build a model for the
IDC with the following growth rates to determine the number affected after
2 weeks:
a) k = .25
b) k=0.25
c) k=.0025
d) k=.00025
2. Consider a spread of rumor concerning termination among 1000
employees of a major company. Assume that the spreading of a rumor is
similar to the spread of contagious disease in that the number hearing the
rumor each day is proportional to the product of those who have heard the
rumor and those who have not heard the rumor. Build a model for the
company with the following rumor growth rates to determine the number
having heard the rumor after 1 week:
a) k = .25
b) k=0.25
c) k=.0025
d) k=.00025
List some ways of controlling the growth rate.
Systems of DDS
Let’s consider systems of difference equations (DDS). For selected initial
conditions, we build numerical solutions to get a sense of long term behavior of
the system. For the systems that we will study, we will find their equilibrium
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
values. We then explore starting values near the equilibrium values to see if the
system
a. remains close,
b. approach the equilibrium value, or
c. not remain close
What happens near these values gives great insight concerning the long-term
behavior of the system. We can study the resulting pattern of the numerical
solutions.
Merchants located Downtown and in Malls
Let's consider the attempt to revitalize the downtown section of a small city
with merchants. There are merchants downtown and others in the large mall.
Suppose historical records determine that 60% of the downtown merchants
remain downtown, while 40% move to the mall. We find the 70% of the mall
merchants want to remain in the mall, but 30% want to move to downtown. Build
a model to determine the long-term behavior of these merchants based upon this
historical data.
Problem Identification: Determine the relationship of the merchants over time.
Assumptions and variables: Let n represent the number of business months.
We define
D(n) = the number of merchants operating downtown at the end of n months
M(n) = the number of merchants operating at the mall at the end of n months
We assume that no other incentives are given to the merchants for either
staying or moving.
The Model:
The number of merchants downtown in any time period is equal to the
number of downtown merchants that stay downtown plus the number of mall
merchants that relocate downtown. The same is true for the number of mall
merchants. In any time period, the number of mall merchants is equal to the
number that remain in the mall plus the number of downtown merchants that
move to the mall. Mathematically, this is written as:
D(n+1) = .6 D(n) + .3 M(n)
M(n+1) = .4 D(n) + .7 M(n)
There are initially 150 merchants in the mall and 100 downtown.
The long-term behavior is found by evaluating graphically and numerically these
equations:
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
The long-term behavior shows that eventually (without other influences) that of
the 250 merchants about 107 merchants will be downtown and 143 will be in the
mall. As a result of our research, we might want to try to attract new businesses
to the community and add incentives for operating their businesses downtown.
Competitive Hunter Models
Competitive hunter models involve species vying for the same resources (such
as food or living space) within their habitat. The effect of the presence of a
second species diminishes the growth rate of the first species. We now consider
a specific example concerning trout and bass in a small pond. Hugh Ketchum
owns a small pond that he uses to stock fish and eventually plans to allow
fishing. He has decided to stock both bass and trout. The fish and game warden
tells Hugh that after inspecting his pond for environmental conditions he has a
solid pond for growth of his fish. In isolation, bass grow at a rate of 20% and trout
at a rate of 30%. The warden tells Hugh that the interactions for the food affect
trout more than bass. They estimate the interaction affecting bass is 0.0010*
bass*trout and for trout is 0.0020* bass*trout. Assume no other changes in the
habitant occur.
Model:
Let
B(n) = the number of bass in the pond after period n.
T(n) = the number of trout in the pond after period n.
B(n) T(n) = interaction of the two species.
B(n+1) = 1.20 B(n) - 0.0010 B(n) T(n)
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
T(n+1) = 1.30 T(n) - 0.0020 B(n) T(n)
Hugh initially buys 151 bass and 199 trout for his pond.
Fast Food Tendencies
Consider your student union center that desires to have three fast food centers
available to students: burgers, tacos, and pizza. These chains run a survey of
students and find the following information concerning lunch: 75% that ate
burgers will eat burgers again at the next lunch, 5% will eat tacos next, and 20%
will eat pizza next. Of those who ate tacos last, 20% will eat burgers next, 60%
will stay will tacos, and 20% will eat pizza next. Of those who ate pizza, 40% will
eat burgers next, 20 % tacos, and 40% pizza again.
We formulate the problem as follows:
Let n represent the nth days lunch and define
B(n) = the number of burger eaters in the nth lunch.
T(n) = the number of taco eaters in the nth lunch.
P(n) = the number of pizza eaters in the nth lunch.
Formulating the system, we have the following dynamical system:
B(n+1) = .75 B(n) + .20 T(n) + .40 P(n)
T(n+1) = .05 B(n) + .60 T(n) + .20 P(n)
P(n+1) = .20 B(n) + .20 T(n) + .40 P(n)
Suppose the campus has 14,000 students that eat lunch.
The results shows that an equilibrium is reached of about 7778 burger eaters,
2722 taco eaters, and 3500 pizza eaters. This allows the fast food
establishments to project the future.
EXERCISES
1. What happens to the merchant problem if 200 merchants were initially in the
mall and 50 were in the downtown?
2. Determine the equilibrium values of the bass and trout. Can these levels ever
be achieved and maintained? Explain.
3. Test the fast food models with different starting conditions summing to 14,000
students. What happens?
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Projects
1. Competitive Hunter Model- A System of DDSs---Spotted Owls and Hawks
PID: Predict the number of owls and hawks in the same environment as a
function of time.
The variables: O(n) = # of owls at then end of period n
H(n) = # of hawks at the end of period n
Model:
O(n+1) = 1.2 O(n) - .001 O(n)H(n)
H(n+1) = 1.3 H(n) - .002 H(n) O(n)
(a) Find the equilibrium values of the system.
(b) In the lab iterate from the following initial conditions and determine what
happens:
Owls
Hawks
150
200
151
199
149
210
10
10
100
100
2. George and Gracie play racket-ball very often and are very competitive. Their
racket-ball match consists of two games. When George wins the first game, he
wins the second game 65% of the time. When Gracie wins the first game, she
wins the second only 45% of the time. Model this as a DDS and determine the
long-term winning percentages of their racket-ball games. What assumptions are
necessary?
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Problem A: Adolescent Pregnancy
You are working temporarily for the Department of Health and Environmental
Control. The director is concerned about the issue of teenage pregnancy in their
region. You have decided that your team will analyze the situation and
determine if it is really a problem in this region. You gather the following 2000
data.
County Age
Age 15- Age 18- 10-14
17
19
births
10-14
15-17
18-19
10-14
births
births
birthsbirthsunmarried unmarried unmarried
281
206
307
184
109
442
201
256
113
446
343
145
437
466
546
326
254
803
345
444
199
686
615
261
16
13
28
15
10
32
7
14
9
22
15
7
Pregnant Pregnant
15-17
birth-
18-19
Pregnant
1
2
3
4
5
6
7
8
9
10
11
12
29
24
40
21
16
44
17
23
13
41
28
9
350
303
422
201
156
523
263
330
123
467
421
179
571
567
691
356
357
970
434
427
221
950
713
311
17
13
29
18
11
33
9
16
10
24
18
8
164
151
251
137
99
293
113
160
78
279
219
114
193
233
366
180
161
396
168
210
106
331
328
162
1998
Age
Pregnancies
Births
10-14
320
231
15-17
4041
3222
18-19
6387
5164
1999
Age
Pregnancies
Births
10-14
309
208
15-17
3882
3048
18-19
6714
5391
Build a mathematical model and use it to determine if there is a problem or not. Prepare an article for the
newspaper that highlights your result in a novel mathematical relationship or comparison that will capture
the attention of the youth.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
ACTIVITY
The following table represents the numbers of sports-related injuries treated in
U.S. hospitals emergency room in 1991, along with an estimate of the number of
participants in that sport.
Sport
Injuries
Participants Sport
Injuries Participants
Basketball
646,678
26,200,000 Fishing
84,115 47,000,000
Bicycling
600,649
54,000,000 Skateboard
56,435 8,000,000
Baseball
459,542
36,100,000 Hockey
54,601 1,800,000
Football
453,684
13,300,000 Golf
38,626 24,700,000
Soccer
150,449
10,000,000 Tennis
29,936 16,700,000
Swimming
130,362
66,200,000 Water skiing 26,663 9,000,000
Weightlifting
86,398
39,200,000 Bowling
25,417 40,400,000
(a)
If we want to use the number of injuries as a measure of the
hazardousness of a sport, which sport is more hazardous between
bicycling and football? Between soccer and hockey?
(b)
use either a calculator or a computer to calculate the rate of injuries per
thousand participants. Rate is defined as the average number of injuries
out of the total participants.
(c)
rank order this new measure for the sports.
(d)
how do your answers in part (a) compare if we do the hazardous analysis
using the rates in (b). If different, why are the results different?
Once you have learned to distinguish between quantitative and categorical data,
we need to move on to a fundamental principle of data analysis: “begin by
looking at a visual display of the data set”.
Global Warming Data Set
Central England
Average
Temperatures
(°C)
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Annual
1961 to 1990
3.8
3.8
5.7
7.9 11.2 14.2 16.1 15.8 13.6 10.6 6.6
4.7
9.5
1880 to 2004
3.8
4.1
5.7
8.0 11.3 14.2 16.1 15.7 13.5 10.0 6.5
4.5
9.5
1659 to 2005
3.2
3.8
5.3
7.9 11.2 14.3 16.0 15.6 13.3 9.7
4.1
9.2
6.0
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Systems of Equations: Chemical Balancing: Yes, HS can do this with
mathematics.
In your chemistry class, you find you are working on an experiment. You find that
chromium compounds exhibit a variety of bright colors. When solid ammonium
dichromate, (NH4)2Cr2O7, a vivid orange compound, is ignited, a spectacular reaction
occurs. Although the reaction is actually quite complex, let’s assume here that the
products are solid chromium (III) oxide, nitrogen gas (consisting of N2 molecules), and
water vapor. The unbalanced equation is:
(NH4)2Cr2O7(s) Cr2O3(s) + N2(g) + H2O(g)
Required:
Using the four chemicals, nitrogen (N), hydrogen (H), chromium (Cr), and oxygen (O)
and multipliers, a, b, c, and d, where:
a (NH4)2Cr2O7(s) b Cr2O3(s) + c N2(g) + d H2O(g)
1) write this as a system of equations and unknowns in matrix form, and
2) solve for the values of a, b, c, and d that balance the
chemical equation.
(NH4)2Cr2O7(s) Cr2O3(s) + N2(g) + H2O(g)
Step 1: Choose a different small letter to multiply in front of each set of terms in the
equation.
a (NH4)2Cr2O7(s) b Cr2O3(s) + c N2(g) + d H2O(g)
a, b, c, and d are these letters
Step 2: Write an equation for each element given and number of atoms used.
Nitrogen (N) : 2a = 2c
Hydrogen (H): 8a = 2d
Chromium (Cr): 2a = 2b
Oxygen (O): 7a = 3b + d
Step 3: Set the equations to 0.
Nitrogen (N) : 2a – 2c = 0
Hydrogen (H): 8a – 2d = 0
Chromium (Cr): 2a – 2b = 0
Oxygen (O): 3b + d – 7a = 0
Step 4: Write as an augmented matrix.
a b c d ans
2 0 -2 0 0
8 0 0 -2 0
2 -2 0 0 0
-7 3 0 1 0
Step 5: With technology, using the reduced row echelon form, solve.
a b c d ans
1 0 0 -.25 0
0 1 0 -.25 0
0 0 1 -.25 0
0000 0
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Step 6: Using the reduced row echelon form, write formulas and set them to 0.
a - .25d = 0
b - .25d = 0
c - .25d = 0
Step 7: Choose any number for d, plug into the formulas, and solve for the remaining
variable.
d=4
a - .25(4) = 0 a = 1
b - .25(4) = 0 b = 1
c - .25(4) = 0 c = 1
Step 8: Write new balanced equation.
(NH4)2Cr2O7(s) Cr2O3(s) + N2(g) + 4H2O(g)
Step 9: Check number of atoms on each side of the equation to see if it is balanced.
Nitrogen (N): 2 2
Hydrogen (H): 8 8
Chromium (Cr): 2 2
Oxygen (O): 7 7
A different balanced equation is:
C2H6O + K2Cr2O7 + H2SO4 C2H4O2 + Cr2(SO4)3 + K2SO4 + H2O
Using the chemicals Carbon (C), Hydrogen (H), Oxygen (O), Sodium (S), Chromium
(Cr), and Potassium (K), set up the multipliers, set up the system of equations, and solve
for the multipliers to balance the system of equations.
Step 1: Choose a different small letter to multiply in front of each set of terms in the
equation.
a C2H6O + b K2Cr2O7 + c H2SO4 d C2H4O2 + Cr2(SO4)3 + e K2SO4 + f H2O
Step 2: Write an equation for each element given and number of atoms used.
Carbon (C): 2a = 2d
Hydrogen (H): 6a +2c = 4d + 2g
Oxygen (O): a + 7b + 4c = 2d + 12e + 4f + g
Potassium (K): 2b = 2f
Chromium (Cr): 2b = 2e
Sodium (S): c = 3e + f
Step 3: Set the equations to 0.
Carbon (C): 2a - 2d = 0
Hydrogen (H): 6a +2c - 4d - 2g = 0
Oxygen (O): a + 7b + 4c - 2d - 12e - 4f – g = 0
Potassium (K): 2b - 2f = 0
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Chromium (Cr): 2b - 2e = 0
Sodium (S): c - 3e - f = 0
Step 4: Write as an augmented matrix.
a b c d e f g ans
2 0 0 -2 0 0 0 0
6 0 2 -4 0 0 -2 0
1 7 4 -2 -12 -4 -1 0
0 2 0 0 0 -2 0 0
0 2 0 0 -2 0 0 0
0 0 1 0 -3 -1 0 0
Step 5: With technology, using the reduced row echelon form, solve in fractional format.
a b c d e f e ans
1 0 0 0 0 0 -3/11 0
0 1 0 0 0 0 -2/11 0
0 0 1 0 0 0 -8/11 0
0 0 0 1 0 0 -3/11 0
0 0 0 0 1 0 -2/11 0
0 0 0 0 0 1 -2/11 0
Step 6: Using the reduced row echelon form, write formulas and set them to 0.
a - 3/11g = 0
b - 2/11g = 0
c - 8/11g = 0
d - 3/11g = 0
e - 2/11g = 0
f - 2/11g = 0
Step 7: Choose any number for g, plug into the formulas, and solve for the remaining
variable.
g = 11
a=3
b=2
c=8
d=3
e=2
f=2
Step 8: Write new balanced equation.
3C2H6O + 2K2Cr2O7 + 8H2SO4 3C2H4O2 + 2Cr2(SO4)3 + 2K2SO4 + 11H2O
Step 9: Check number of atoms on each side of the equation to see if it is balanced.
It is!
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
HiMCM Problem B: Forest Service
Your team has been approached by the Forest Service to help allocate resources to fight
wildfires.
In particular, the Forest Service is concerned about wildfires in a wilderness area
consisting of small trees and brush in a park shaped like a square with dimensions 80 km
on a side. Several years ago, the Forest Service constructed a network of north-south and
east-west firebreaks that form a rectangular grid across the interior of the entire
wilderness area. The firebreaks were built at 5 km intervals.
Wildfires are most likely to occur during the dry season, which extends from July
through September in this particular region. During this season, there is a prevailing
westerly wind throughout the day. There are frequent lightning bursts that cause
wildfires.
The Forest Service wants to deploy four fire-fighting units to control fires during the next
dry season. Each unit consists of 10 firefighters, one pickup truck, one dump truck, one
water truck (50,000 liters), and one bulldozer (w/ truck and trailer). The unit has
chainsaws, hand tools, and other fire-fighting equipment. The people can be quickly
moved by helicopter within the wilderness area, but all the equipment must be driven via
the existing firebreaks. One helicopter is on standby at all times throughout
the dry season.
Your task is to determine the best distribution of fire-fighting units within the wilderness
area. The Forest Service is able to set up base camps for those units at sites anywhere
within the area. In addition, you are asked to prepare a damage assessment forecast.
This forecast will be used to estimate the amount of wilderness likely to be burned by fire
as well as acting as a mechanism for helping the Service determine when additional fire-fighting
units need to be brought in from elsewhere.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Linear Programming – Helping Victims of a Disaster
Packages of food and clothing are being sent to assist victims in a disaster.
Carriers will transport the packages, provided they fit in the available cargo
space. Each 20-cubic ft. box of food weighs 40 lbs., and each 30-cubic. ft.
box of clothing weighs 20lbs. The total weight cannot exceed 16,000 lbs.,
and the total volume must not exceed 18,000- cubic ft. Each carton of food
will feed 10 people, while each carton of clothing will help put clothes on 8
people. How many packages of food and how many packages of clothing
should be sent in order to maximize the number of people assisted? How
many people will be assisted?
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
The Terror Bird
No, we didn't make this up! Titanis really lived &— about 2 million years ago on
the oak and grass Savannahs of what is now Florida. Dr. Bob Chandler, one of the
world's experts on fossil birds, now at Georgia College and State University, has dredged
up a number of bits and pieces of this bizarre predatory bird from the Santa Fe River near
Gainesville, Florida.
 Giant, flightless predatory bird
 Lived in South America 30 million years before the Interchange
 Fossils have been found of the Terror Bird, Titanis Walleri, in Florida
 Suspected to be a fierce hunter who would lie and ambush its prey and attack from the
tall grasslands.
 Suspected to pin down its prey with its beak.
 Suspected to use 4-5 inch inner toe claw with its beak to shred its prey
 Another unique feature was this bird had arms not wings (more powerful than the arms
of the Velociraptor (Chandler, 1994)--about the size of an NFL linebacker.
 Bones give a good indication of size (height) but not reliable body weight (weight is not
easily fossilized)
 We want to infer its weight.
Web Sites: Try a “google search” on the Terror Bird
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Problem Identification Statement: Predict the size of the terror bird found as a function
of the circumference of its femur.
Assumptions and Variables: The terror bird was a prehistoric bird. Therefore, it is
similar to a modern bird as well as it was similar to dinosaurs of its time. The Terror
Bird's femur is 21 cm in circumference.
The following data is initially available:
Bird Data
Femur circumference (cm)
.7943
.7079
1.000
1.122
1.6982
1.2023
1.9953
2.2387
2.5119
2.5119
3.1623
3.5481
4.4668
5.8884
6.7608
15.136
15.850
Weight (kg)
.0832
.0912
.1413
.1479
.2455
.2818
.7943
2.5119
1.4125
.8913
1.9953
4.2658
6.3096
11.2202
19.9500
141.25
158.4893
Steps in the modeling process:
a) Proportionality & Geometric Similarity
We build a model based on proportionality and geometric similarity arguments (GS)
a) Terror bird is a scale model of other birds
b) Volume is proportional to length3
c) Constant density ad constant gravity show Volume proportional to weight
(W=mg)
d) W proportional to l3
3
W = kl
1) Plot the raw data. Discuss or comment on the shape of the data. Is the data
increasing or decreasing? Is it concave up or concave down? Is it linear or non-linear?
2) Transform the data for length to length3.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
3) Plot the transformed data, W vs. l3. If it appears like the data is a straight line
through the origin then we have our support for the proportionality argument.
4) Using model fitting (least squares) find the best fit line.
5) Plot the real data and the curve together.
6) Plot the residuals and look for trends.
B.) EMPIRICAL Modeling
1) Plot the raw data. Discuss or comment on the shape of the data. Is the data
increasing or decreasing? Is it concave up or concave down? Is it linear or non-linear?
2) What function fits this curve? (Our ability to model curves is not good but we can
model lines.)
3) How can we make this data linear? We can take the natural logarithm of each data
element. This transformation might linearize the data and make it easier to model.
4) Plot the transformed data, ln weight versus ln femur circumference. Does it appear
linear?
5) Since it appear linear and depending on what you have covered in your courses then:
a) Pick two points that fall on your line and find the slope and the intercept, or
b) To model linear regression, use the transformed data and solve the system of
equation for the slope,m , and the intercept b, or
m  (ln x)2 + b  (ln x) =  (ln x ln y)
m  (ln x) + b n
=  ( ln y)
c) Use technology and linear regression to find the "best" fit slope and intercept.
6) Using the laws of logs and exponential models, transform the linear model,
ln y = m ln x + ln b, into
e ln y  (e ln b )(e ln x )
m
y  (e ln b )( x m )
7) Check to see how "reasonable" your model really is. Check the absolute error |ya-yp|.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Extensions and modifications:
1) Use dinosaur data instead of bird data.
Dinosaur Data
Name
Femur Circumference (mm) Weight (kg)
Hypsilophodontidae
103
55
Ornithomimdae
136
115
Thescelosauridae
201
311
Ceratosauridae
267
640
Allosauridae
348
1230
Hadrosauridae-1
400
1818
Hadrosauridae-2
504
3300
Hadrosauridae-3
512
3500
Tyrannosauridae
534
4000
2) Compare and contrast the result from the two data sets. The answers are different.
3) Discuss domain and range issues of data, interpolation versus extrapolation.
4) Check for a directly proportional argument, like W l3.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Modeling Problem
Consider the following problem: CA State's Water Commission is requiring data from La Mesa
housing on the rate of water use, in gallons per hour, and the total amount of water used each day.
The Department of Engineering and Housing (DEH) in Monterey does not have the sophisticated
equipment that measures the flow of water in or out of the main water tank. Instead, DEH can
measure only the level of water in the tank, within 0.5% accuracy, every hour. More importantly,
whenever the level in the tank drops below some minimum level L, a pump fills the tank up to the
maximum level, H, but there is no measurement of the pump flow at these times either. Thus,
one cannot readily relate the level in the tank to the amount of water used while the pump is
working, which occurs once or twice per day, for a couple of hours each time. The table below
contains the time, in seconds, since the first measurement, and the level of water in the tank in
hundredths of a foot. For example, after 3316 seconds, the depth of the water in the tank reached
31.10 feet. The tank is a vertical circular cylinder with a height of 40 feet and a diameter of 57
feet. Usually, the pump starts filling the tank when the level drops to about 27.00 feet and the
pump stops when the level rises back to about 35.50 feet.
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Time (sec)
Level (.01 ft)
0
3175
3316
3110
6635
3054
10619
2994
13937
2947
17921
2892
21240
2850
25223
2795
28543
2752
32884
2697
35932
pump on
39332
pump on
39435
3550
43318
3445
46636
3350
49953
3260
53936
3167
57254
3087
60574
3012
64554
2927
68535
2842
71854
2757
75021
2697
79254
pump on
82649
pump on
85968
3475
89953
3397
93270
3340
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
CO2 Emissions
Year
CO2
(metric
tons)
1960
16.2
1961
16.3
1962
16.7
1963
17.2
1964
17.7
1965
18.4
1966
18.9
1967
19.5
1968
20.2
1969
20.4
1970
20.6
1971
20.4
1972
21.1
1973
21.7
1974
20.8
1975
19.7
1976
20.8
1977
20.8
1978
21.4
1979
21.2
1980
20.3
1981
19.3
1982
18.2
1983
18.2
1984
18.6
1985
18.4
1986
19
1987
19.6
1988
19.8
1989
19.2
1990
19
1991
18.8
1992
19.7
1993
19.8
1994
19.5
1995
19.7
1996
20
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
1997
19.5
1998
19.6
1999
20
2000
19.6
2001
19.6
2002
19.4
2003
19.5
2004
19.5
CO2 Emissions
25
Metric Tons, CO2
20
15
Series1
10
5
0
1950
1960
1970
1980
Year
1990
2000
2010
Mathematical Modeling for Montana Green Technology Projects
9-11 April 2010
Dr. Bill Fox & Dr. Frank Giordano
Problem: Going Green
The United States can address its national carbon footprint in two ways: by reducing carbon dioxide emissions or by
increasing carbon dioxide consumption (sequestration). Assume that the total U.S. carbon dioxide emissions are capped
at 2007-2008 levels indefinitely. What should the U.S. do to increase carbon dioxide consumption to achieve national
carbon neutrality with minimal economic and cultural impact? Is it even possible to achieve neutrality? Model your
solution to show feasibility, effectiveness, and costs. Prepare a short summary paper for the U.S. Congress to persuade
them to adopt your plan.
Problem: Water, Water Everywhere
Fresh water is the limiting constraint for development in much of the United States. Devise an effective, feasible, and
cost-efficient national water strategy for 2010 to meet the projected needs of the United States in 2025. In particular,
address storage and movement, de-salinization, and conservation as some of the possible components of your
strategy. Consider economic, physical, cultural, and environmental effects. Provide a position paper for the United
States Congress outlining your approach, its costs, and why it is the best choice for the nation.
Problem B
Problem: Tsunami ("Wipe Out!")
Recent events have reminded us about the devastating effects of distant or underwater earthquakes. Build a model
that compares the devastation of various-sized earthquakes and their resulting Tsunamis on the following cities: San
Francisco, CA; Hilo, HI; New Orleans, LA; Charleston, SC; New York, NY; Boston, MA; and any city of your
choice. Prepare an article for the local newspaper that explains what you discovered in your model about one of these
cities.