College Algebra (MA134-15) Honors Projects



College Algebra (MA134-10H) Honors Projects

Version 8/29/2007

Project 1. Cell Phone Service.

Based on Michael Sullivan, College Algebra, 7th Edition (Pearson Prentice Hall 2005)

This project requires a graphing calculator and some knowledge about functions and their graphs.

In purchasing cell phone service, you must shop around to determine the best rates and services that apply to your situation. Suppose that you want to upgrade your old model cell phone. Yu have gathered the following information on cell phone services:

▪ Nokia 5156 $19.99 for the phone

▪ Ericsson A1228di Free (after $20 rebate)

▪ Option 1: $29.99 for 250 peak minutes, unlimited nights and weekends; extra peak time minutes cost $0.45 per minute

▪ Option 2: $39.99 for 400 peak minutes, unlimited nights and weekends; extra peak time minutes cost $0.45 per minute

▪ Option 3: $49.99 for 600 peak minutes, unlimited nights and weekends; extra peak time minutes cost $0.35 per minute

To qualify for these phone prices, you must sign up for a 2-year contract.

a) List the different combinations of phone and services.

b) Determine the total cost of each combination described in part (a) for the life of the contract (24 months), assuming that you will stay within the allotted peak time minutes provided by each contract.

c) If you expect to use 260 peak time minutes per month, which option provides the best deal? If you expect to use 320 peak time minutes per month, which option provides the best deal?

d) If you expect to use 410 peak time minutes per month, which option provides the best deal? If you expect to use 450 peak time minutes per month, which option provides the best deal?

e) Each monthly charge includes a specific number of peak time minutes included in the monthly fee. Write a function for each available option, where C is the monthly cost and x is the number of peak time minutes used.

f) Graph the functions corresponding to each option.

g) At what point does Option 2 become a better deal than Option 1?

h) At what point does Option 3 become a better deal than Option 2?

This project is based on information given in a “Cingular Wireless” advertisement in the Denton Record Chronicle on September 11, 2001

Project 2. Dynamical Systems, Spotted Owls, Blue Whales, Rabbits and Wolves.

Based on David C. Lay, Linear Algebra and Its Applications, 2nd Edition (Addison-Wesley, 1997) and TI92 manual. This project will use MS Excel.

In this case study, we examine how recursive sequences can be used to study the change in a population over time.

The population of spotted owls is divided into three age classes: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). The population is examined at yearly intervals. Since it is assumed that the number of male and female owls is equal, only female owls are counted in the analysis. If there are jk juvenile females, sk subadult females, and ak adult females at year k, then R. Lamberson et al. (see Reference 3) found that the population of owls could be modeled by the equations followed by initial populations:

jk+1 = 0.33ak j0 = 40

sk+1 = 0.18 jk s0 = 20

ak+1 = 0.71 sk + 0.94 ak a0 =140

The entries in the first row describe the fecundity of the population. Thus in the model above juveniles and subadults do not produce offspring, but each adult female produces (on the average) .33 juvenile females per year. The other entries in the matrix show survival. In this model, 18% of the juvenile females survive to become subadults, 71% of the subadults survive to become adults, and 94% of the adults survive each year. Note that the measures of fecundity and survival remain constant through time.

We wish to determine the long-term dynamics of the population: whether the population is becoming extinct or is increasing. To answer these questions we will use Excel to produce recursive sequences (jk), (sk), (ak) and examine the population of juvenile, subadult, and adult females and the total population of spotted owls over the period of 100 years. Based on these observations we will make some prediction regarding the future of the species: extinction or survival. If the population increases, we will find the long-term distribution of the owls by life stages.

1. Determine the future of spotted owls modeled by the equations above. Will the species survive or become extinct in the long term future?

2. If the survival rate for juveniles were somehow increased to 30%, how would this reflect on the future of the species? What percentage of the total population of females would juvenile, subadult, and adult females be?

3. The data available in reference 4 gives the following survival and fecundity rates:

Juvenile Survival .33

Subadult Survival .85

Adult Survival .85

Subadult Fecundity .125

Adult Fecundity .26

Using this data, determine the long-range population of the Northern spotted owl. Are prospects for the owl better or worse than given in the data in the example above?

4. Blue Whales

In the 1930's (before its virtual extinction and a great change in its survival rates) a researcher studied the blue whale population (see References 2, 5 and 6 for this data). Due to the long gestation period, mating habits, and migration of the blue whale, a female can produce a calf only once in a two-year period. Thus the age classes for the whale were assumed to be: less than 2 years (ak), 2 or 3 years (bk), 4 or 5 years (ck), 6 or 7 years (dk), 8 or 9 years (ek), 10 or 11 years (fk), and 12 or more years (gk). The model for survival and fecundity is given by:

ak+1 = 0.19ck + 0.44dk + 0.50ek + 0.50fk + 0.45gk a0 = 50

bk+1 = 0.77ak b0 = 35

ck+1 = 0.77bk c0 = 25

dk+1 = 0.77ck d0 = 20

ek+1 = 0.77dk e0 = 10

fk+1 = 0.77ek f0 = 10

gk+1 = 0.77fk + 0.78gk g0 = 40

Determine whether the blue whale population is becoming extinct in this model. If the population is not becoming extinct, determine the percentage of each class in the stable population.

5. Rabbits and Wolves

This is a simplified example of a predator-prey model. We will use it to determine the number of rabbits and wolves that maintain population equilibrium in a certain region. We will assume that there is enough food for rabbits at any time.

rn = number of rabbits

m = growth rate of rabbits if there are no wolves (use 0.05)

k = rate at which wolves kill rabbits (use 0.001)

wn = number of wolves

g = growth rave of wolves if there are rabbits to eat (use 0.0002)

d = death rate of wolves if there are no rabbits to eat (use 0.03)

Use the initial population values r0 = 200 and w0 = 50 and run the simulation for 500 cycles to determine what happens to rabbits and wolves. Create two plots: population of both species vs. time and population of wolves vs. population of rabbits. Change r0 and w0 . Do you observe any changes? What if you change the rates m, k, g, d?

References:

1. Huenneke, L. F. and Marks, P. L. “Stem Dynamics of the Shrub Alcus Incana SSP. Rugosa: Transition Models.” Ecology 68 (1987), 1234-1242.

2. Jffers, John N. R. An Introduction to Systems Analysis: with Ecological Applications. London: Edward Arnold, 1978.

3. Lamberson, R. H. et al. “A Dynamic Analysis of the Viability of the Northern Spotted Owl in a Fragmented Forest Environment.” Conservation Biology 6(1992), 505-512.

4. Lamberson, R. H. Private communication, 1999.

5. Laws, R. M. “Some E_ects of Whaling on the Southern Stocks of Baleen Whales.” In The Exploitation of Natural Animal Populations, 242-259. Oxford: Blackwells, 1962.

6. Usher, M. B. “Developments in the Leslie Matrix Model.” In Mathematical Models in Ecology, 29-60. Oxford: Blackwells, 1972.

7. TI-92 guidebook

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