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a first course in mathematical modeling

Questions and Answers of A First Course In Mathematical Modeling

A new planet is discovered beyond Pluto at a mean distance to the sun of 4004 million miles. Using Kepler's third law, determine an estimate for the time T to travel around the sun in an orbit.
People are fixated by speed. Which computer systems offer the most speed?For the scenarios presented in Problems 9-17, identify a problem worth studying and list the variables that affect the
A company with a fleet of trucks faces increasing maintenance costs as the age and mileage of the trucks increase.For the scenarios presented in Problems 9-17, identify a problem worth studying and
A physicist is interested in studying properties of light. He wants to understand the path of a ray of light as it travels through the air into a smooth lake, particularly at the interface of the two
Animals of different sizes work differently. Small ones have squeaky voices, their hearts beat faster, and they breathe more often than larger ones. On the other hand, the skeleton of a larger animal
A botanist is interested in studying the shapes of leaves and the forces that mold them. She clips some leaves from the bottom of a white oak tree and finds the leaves to be rather broad and not very
In what circumstances, if any, will Models (2.11) and (2.13) coincide? Explain fully.
Again, consider the spreading of a rumor (see Problem 13 in this section), but now assume a company with 2000 employees. The rumor concerns the number of mandatory terminations that this company must
Explain the meaning of the following proportionality argument and illustrate it graphically. κα -2 (1)
A transportation company is considering transporting people between skyscrapers in New York City via helicopter. You are hired as a consultant to determine the number of helicopters needed. Identify
You are considering a 30-year mortgage that charges 0.4% interest each month to pay off a $250,000 mortgage.a. Determine the monthly payment p that allows the loan to be paid off at 360 months.b.
The following data represent the U.S. population from 1790 to 2010. Find a dynamical system model that fits the data fairly well. Test your model by plotting the predictions of the model against the
Your parents are considering a 30-year, $100,000 mortgage that charges 0.5% interest each month. Formulate a model in terms of a monthly payment p that allows the mortgage (loan) to be paid off after
Name several behaviors you think can be modeled by dynamical systems.
You currently have $5000 in a savings account that pays 0.5% interest each month. You add $200 each month. Build a numerical solution to determine when the account reaches $20,000.
Assume that we are considering the survival of whales and that if the number of whales falls below a minimum survival level m, the species will become extinct. Assume also that the population is
For the following problems, find the solution to the difference equation and the equilibrium value if one exists. Discuss the long-term behavior of the solution for various initial values. Classify
Consider the spreading of a highly communicable disease on an isolated island with population size N. A portion of the population travels abroad and returns to the island infected with the disease.
Sociologists recognize a phenomenon called social diffusion, which is the spreading of a piece of information, a technological innovation, or a cultural fad among a population. The members of the
Complete the modules ``The Growth of Partisan Support I: Model and Estimation''(UMAP 304) and ``The Growth of Partisan Support II: Model Analytics'' (UMAP 305), by CarolWeitzelKohfeld.UMAP304
Is college a financially sound investment? Income is forfeited for 4 years, and the cost of college is extremely high. What factors determine the total cost of a college education? How would you
How fast can a skier ski down a mountain slope?In Problems 1-8, the scenarios are vaguely stated. From these vague scenarios, identify a problem you would like to study. Which variables affect the
How far can a person long jump? In the 1968 Olympic Games in Mexico City, Bob Beamon of the United States increased the record by a remarkable 10%, a record that stood through the 1996 Olympics. List
The United States Food and Drug Administration is interested in knowing if a new drug is effective in the control of a certain disease in the population.In Problems 1-8, the scenarios are vaguely
Consider Models (2.11) and (2.13). Which do you think is better? Why? Discuss the models qualitatively. In Chapter 3 you will be asked to compare the two models analytically.
Determine whether the following data support a proportionality argument for y ∝ z1/2. If so, estimate the slope. y N 3.5 5 6 7 8 3 6 9 12 15
Scaling a Braking System— Suppose that after years of experience, your auto company has designed an optimum braking system for its prestigious full-sized car. That is, the distance required to
When should a person replace his or her vehicle? What factors should affect the decision? Which variables might be neglected initially? Identify the data you would like to have to determine the
How should a manufacturer of some products decide how many units of that product should be manufactured each year and how much to charge for each unit?In Problems 1-8, the scenarios are vaguely
A road map has scale 1 inch = 6 miles. You measure the distance from home to the ski resort you plan to go visit as 11.75 inches. How many miles will you be traveling?What assumptions are you making?
Racing Shells — If you have been to a rowing regatta, you may have observed that the more oarsmen there are in a boat, the faster the boat travels. Investigate whether there is a mathematical
Consider the operation of a medical office. Records have to be kept on individual patients, and accounting procedures are a daily task. Should the office buy or lease a small computer system? Suggest
An object is to be dropped from a great height. When and how hard will it hit the ground?In Problems 1-8, the scenarios are vaguely stated. From these vague scenarios, identify a problem you would
Consider the situation of two warm-blooded adult animals essentially at rest and under the same conditions (as in a zoo). Assume that the animals maintain the same body temperature and that the
If an architectural drawing is scaled so that 0.75 in. represents 4 ft, what length represents 27 ft?
Lumber Cutters — Lumber cutters wish to use readily available measurements to estimate the number of board feet of lumber in a tree. Assume they measure the diameter of the tree in inches at waist
Should a couple buy or rent a home? As the cost of a mortgage rises, intuitively, it would seem that there is a point where it no longer pays to buy a house. What variables determine the total cost
How would you design a lecture hall for a large class?In Problems 1-8, the scenarios are vaguely stated. From these vague scenarios, identify a problem you would like to study. Which variables affect
Assume that under certain conditions the heat loss of an object is proportional to the exposed surface area. Relate the heat loss of a cubic object with side length 6 in. to one with a side length of
It has been said that gymnastics is a sport that requires great agility. Use the model and assumptions developed for agility in this section to argue why there are few tall gymnasts.
If a spring is stretched 0.37 in. by a 14-lb force, what stretch will be produced by a 9-lb force? By a 22-lb force? Assume Hooke's law, which asserts that the distance stretched is proportional to
Heart Rate of MammalsThe following data relate the weights of some mammals to their heart rate in beats per minute. Based on the discussion relating blood flow through the heart to body weight, as
Consider wine making. Suggest some objectives a commercial producer might have. Consider taste as a submodel. What are some of the variables affecting taste? Which variables might be neglected
A farmer wants to maximize the yield of a certain crop of food grown on his land. Has the farmer identified the correct problem? Discuss alternative objectives.In Problems 1-8, the scenarios are
A popular measure of physical condition and personal appearance is the pinch test. To administer this test, you measure the thickness of the outer core at selected locations on the body by pinching.
Discuss several factors that were completely ignored in our analysis of the gasoline mileage problem.
Heart Rate of BirdsWarm-blooded animals use large quantities of energy to maintain body temperature because of the heat loss through the body surface. In fact, biologists believe that the primary
A retail store intends to construct a new parking lot. How should the lot be illuminated?In Problems 1-8, the scenarios are vaguely stated. From these vague scenarios, identify a problem you would
Consider a 20-lb pink flamingo that stands 3 ft in height and has legs that are 2 ft in length. Model the height and leg length of a 100-lb flamingo. What assumptions are necessary? Are they
Research and prepare a 10-minute report on Hooke's law.
Tests exist to measure the percentage of body fat. Assume that such tests are accurate and that a great many carefully collected data are available. You may specify any other statistics, such as
Show graphically the meaning of the proportionality y ∝ u/v.
In the automobile gasoline mileage example, assume the drag forces are proportional to Sv, where S is the cross-sectional area perpendicular to the direction of the moving car and v is its speed.
SuperstarsIn the TV show Superstars the top athletes from various sports compete against one another in a variety of events. The athletes vary considerably in height and weight. To compensate for
Consider the taste of brewed coffee. What are some of the variables affecting taste? Which variables might be neglected initially? Suppose you hold all variables fixed except water temperature. Most
The population growth of a single species.In Problems 1-8, the scenarios are vaguely stated. From these vague scenarios, identify a problem you would like to study. Which variables affect the
Suppose two maps of the same country on different scales are drawn on tracing paper and superimposed, with possibly one of the maps being turned over before being superimposed on the other. Show that
Consider an endurance test that measures only aerobic fitness. This test could be a swimming test, running test, or bike test. Assume that we want all competitors to do an equal amount of work.
Consider an automobile suspension system. Build a model that relates the stretch (or compression) of the spring to the mass it supports. If possible, obtain a car spring and collect data by measuring
In the automobile gasoline mileage example, suppose you plot miles per gallon against speed as a graphical representation of the general rule (instead of the graph depicted in Figure 2.27). Explain
Consider modeling the contagious disease Ebola. (You may be interested in researching on the Internet just how deadly this virus is.) An animal research laboratory is located in Restin, Virginia, a
Consider the spreading of a rumor through a company of 1000 employees, all working in the same building. We assume that the spreading of a rumor is similar to the spreading of a contagious disease
Your current credit card balance is $12,000 with a current rate of 19.9% per year. Interest is charged monthly. Determine what monthly payment p will pay off the card in:a. Two years, assuming no
Continuation of Problem 8, Section 1.2: Determine the age of the humanoid skull found near the remains of an ancient campfire.Data from problem 8Ahumanoid skull is discovered near the remains of an
Cipro is an antibiotic taken to combat many infections, including anthrax. Cipro is ltered from the blood by the kidneys. Each 24-hour period, the kidneys lter out about one third of the Cipro that
Repeat Problem 10 with an interest rate of 0.5%. For Problems 7–10, formulate a dynamical system that models change exactly for the described situation.Data from problem 10Your grandparents have an
Continuation of Problem 6, Section 1.2: Experiment with different initial and maintenance doses. Find a combination that is convenient, considering the time between doses and the amount that will be
In model (1.11) determine the outcome with the following parameters changed:a. Initially 5 are sick, and 15 are sick the next week.b. The flu lasts 1 week.c. The flu lasts 4 weeks.d. There are
Place a cold can of soda in a room. Measure the temperature of the room, and periodically measure the temperature of the soda. Formulate a model to predict the change in the temperature of the soda.
Your grandparents have an annuity. The value of the annuity increases each month as 1% interest on the previous month's balance is deposited. Your grandparents withdraw $1000 each month for living
The data in the accompanying table show the speed n (in increments of 5 mph) of an automobile and the associated distance an in feet required to stop it once the brakes are applied. For instance, n =
Your parents are considering a 30-year, $200,000 mortgage that charges 0.5% interest each month. Formulate a model in terms of a monthly payment p that allows the mortgage (loan) to be paid off after
Your parents are considering a 30-year mortgage that charges 0.5% interest each month.Formulate a model in terms of a monthly payment p that allows the mortgage (loan) to be paid off after 360
Reexamine the airlines' tendencies at the regional airport adding the assumption that each period, 100 new passengers enter the system. Experiment with changing the distribution of the passengers.
Ahumanoid skull is discovered near the remains of an ancient campfire. Archaeologists are convinced the skull is the same age as the original campre. It is determined from laboratory testing that
You owe $500 on a credit card that charges 1.5% interest each month. You pay $50 each month and you make no new charges. For Problems 7 –10, formulate a dynamical system that models change exactly
In 1868, the accidental introduction into the United States of the cottony-cushion insect (Icerya purchasi) from Australia threatened to destroy the American citrus industry.To counteract this
Use the model developed in Problem 6 to prescribe an initial dosage and a maintenance dosage that keeps the concentration above the effective level of 500 ppm but below a safe level of 1000 ppm.
You currently have $5000 in a savings account that pays 0.5% interest each month. You add another $200 each month. For Problems 7–10, formulate a dynamical system that models change exactly for
You owe $500 on a credit card that charges 1.5% interest each month. You can pay $50 each month with no new charges. What is the equilibrium value? What does the equilibrium value mean in terms of
A certain drug is effective in treating a disease if the concentration remains above 100 mg/L. The initial concentration is 640 mg/L. It is known from laboratory experiments that the drug decays at
In Example 4, Travelers' Tendencies at a Regional Airport, experiment with starting values near the origin. Does the origin appear to be a stable equilibrium? Explain.Experiment with different values
Complete the UMAP module ``Difference Equations with Applications,'' by Donald R. Sherbert, UMAP 322. This module presents a good introduction to solving first and second order linear difference
Mercury in Fish—Public officials are worried about the elevated levels of toxic mercury pollution in the reservoirs that provide the drinking water to your city. They have asked for your assistance
Write out the first five terms of the sequence satisfying the following difference equations:a. b. c. d. Δαn = jan, do = 1
Build a numerical solution for the following initial value problems. Plot your data to observe patterns in the solution. Is there an equilibrium solution? Is it stable or unstable?a. b. c. d. e.
In analyzing the battle of Trafalgar in 1805, we saw that if the two forces simply engaged head-on, the British lost the battle and approximately 24 ships, whereas the French Spanish force lost
A Killer Virus You have volunteered for the Peace Corps and have been sent to Rwanda to help in humanitarian aid. You meet with the World Health Organization (WHO) and find out about a new killer
For the following problems, find an equilibrium value if one exists. Classify the equilibrium value as stable or unstable.a. b. c. d. e. f.g. h. i. j. k. an+1 = 1.1an
Assume we are considering the survival of whales and that if the number of whales falls below a minimum survival level m, the species will become extinct. Assume also that the population is limited
Prepare a summary of the paper by May et al. listed among the Further Reading titles for this section.
Find a formula for the nth term of the sequence.a. b. c. d. (3,3, 3, 3, 3,...}
The following data were obtained for the growth of a sheep population introduced into a new environment on the island of Tasmania.1Plot the data. Is there a trend? Plot the change in population
With the price of gas continuing to rise, you wish to look at cars that get better gas mileage. You narrow down your choices to the following 2012 models: Ford Fiesta, Ford Focus, Chevy Volt, Chevy
You plan to invest part of your paycheck to nance your children's education. You want to have enough in the account to draw $1000 a month every month for 8 years beginning 20 years from now. The
Find the solution to the difference equations in the following problems:a. b. c. d. e. f. an+1 -Зап, заn, ao = 1
Complete the UMAP module ``The Diffusion of Innovation in Family Planning,'' by Kathryn N. Harmon, UMAP 303. This module gives an interesting application of finite difference equations to study the
Complete the requirements of the UMAP module ``Graphical Analysis of Some Difference Equations in Biology,'' by Martin Eisen, UMAP 553. The growth of many biological populations can be modeled by

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