Questions and Answers of A First Course In Mathematical Modeling

Complete the requirement for UMAP module 269, ``Monte Carlo: The Use of Random Digits to Simulate Experiments,'' by Dale T. Hoffman. The Monte Carlo technique is presented, explained, and used to
Using Monte Carlo simulation, write an algorithm to calculate the volume trapped between the two paraboloids:
Using Monte Carlo simulation, write an algorithm to calculate that part of the volume of an ellipsoid:
Find the area trapped between the two curves y = x2 and y = 6 –x and the x- and y-axes.
Use Monte Carlo simulation to approximate the area under the curve f (x) = √x, over the interval 1/2 ≤x ≤ 3/2.
Using Monte Carlo simulation, write an algorithm to calculate an approximation to π by considering the number of random points selected inside the quarter circle:
Two record companies, A and B, produce classical music recordings. Label A is a budget label, and 5% of A's new compact discs exhibit significant degrees of warpage. Label B is manufactured under
Each ticket in a lottery contains a single ``hidden'' number according to the following scheme: 55% of the tickets contain a 1, 35% contain a 2, and 10% contain a 3. A participant in the lottery wins
You can use the cubic spline software you developed, coupled with some graphics, to draw smooth curves to represent a figure you wish to draw on the computer. Overlay a piece of graph paper on a
The following data represent the weight-lifting results from the 1976 Olympics.For Projects 2-9, use the software you developed in Project 1 to find the splines that pass through the given data
The following data represent the length and weight of a fish (bass).For Projects 2-9, use the software you developed in Project 1 to find the splines that pass through the given data points. Use
The following data represent the pace of life (see Problem 1, Section 4.1). P is the population and V is the mean velocity in feet per second over a 50-ft course.For Projects 2-9, use the software
The following data represent the population of the United States from 1790 to 2000.For Projects 2-9, use the software you developed in Project 1 to find the splines that pass through the given data
The data presented in Table 4.22 on the growth of yeast in a culture (data from R. Pearl, ``The Growth of Population,'' Quart. Rev. Biol. 2(1927): 532548).Table 4.22For Projects 2-9, use the software
The Cost of a Postage StampConsider the following data. Use the procedures in this chapter to capture the trend of the data if one exists. Would you eliminate any data points? Why? Would you be
For Problems 2 and 3, find the natural cubic splines that pass through the given data points. Use the splines to answer the requirements.a. Estimate the derivative evaluated at x = 3.45. Compare
The following data represent the weight-lifting results from the 1976 Olympics.In Problems 6-12, construct a scatterplot of the given data. Is there a trend in the data? Are any of the data points
The following data represent the length of a bass fish and its weight.In Problems 6-12, construct a scatterplot of the given data. Is there a trend in the data? Are any of the data points outliers?
The following data represent the pace of life'' data (see Problem 1, Section 4.1). P is the population and V is the mean velocity in feet per second over a 50-ft course.In Problems 6-12,
The following data were obtained for the growth of a sheep population introduced into a new environment on the island of Tasmania. (Adapted from J. Davidson, ``On the Growth of the Sheep Population
The following data represent the population of the United States from 1790 to 2000.In Problems 6-12, construct a scatterplot of the given data. Is there a trend in the data?Are any of the data points
Construct a scatterplot for the ``yeast growth in a culture'' data. Do the data seem reasonable? Construct a divided difference table. Try smoothing with a low-order cubic polynomial using an
In the following data, X is the Fahrenheit temperature and Y is the number of times a cricket chirps in 1 minute (see Problem 7, Section 4.1). Make a scatterplot of the data and discuss the
For the tape recorder problem in this section, give a system of equations determining the coefficients of a polynomial that passes through each of the data points. If a computer is available,
The following data give the population of the United States from 1800 to 2000. Model the population (in thousands) as a function of the year. How well does your model fit? Is a one-term model
The following data represent the length and weight of a set of fish (bass). Model weight as a function of the length of the fish.
The following data measure two characteristics of a ponderosa pine. The variable X is the diameter of the tree, in inches, measured at breast height; Y is a measure of volume–the number of board
Fit a model to Table 4.9. Do you recognize the data? What relationship can be inferred from them?Table 4.9
In Table 4.8, X is the Fahrenheit temperature, and Y is the number of times a cricket chirp in 1 minute. Fit a model to these data. Analyze how well it fits.Table 4.8
Graph the equation you found in Problem 1f superimposed on the original scatterplot.Data from problem 1fFit the model V = CPa to the ``pace of life'' data in Table 4.5. Use the transformation log V =
Fit the model V = CPa to the ``pace of life'' data in Table 4.5. Use the transformation log V = a log P + log C. Plot log V versus log P. Does the relationship seem reasonable?Table 4.5b. Construct a
Problem 2 in Section 3.1Data from problem 2The following table gives the elongation e in inches per inch (in./in.) for a given stress S on a steel wire measured in pounds per square inch (lb/in.2).
Problem 5a in Section 3.1Data from problem 5aThe following data represent the growth of a population of fruit flies over a 6-week period. Test the following models by plotting an appropriate set of
Problem 4b in Section 3.1Data from problem 4bIn the following data, V represents a mean walking velocity and P represents the population size. We wish to know if we can predict the population size P
Data for planets:Fit the model y = ax3/2.In Problems 7–10, fit the data with the models given, using least squares.
Make an appropriate transformation to fit the model P = aebt using Equation (3.4). Estimate a and b.equation 3.4
For the following data, formulate the mathematical model that minimizes the largest deviation between the data and the model P = aebt . If a computer is available, solve for the estimates of a and b.
The following data represent (hypothetical) energy consumption normalized to the year 1900. Plot the data. Test the model Q= aebx by plotting the transformed data. Estimate the parameters of the
Data have been collected on numerous dinosaurs during the prehistoric period. Using proportionality and geometric similarity, build a mathematical model to relate the weight of the terror bird to its
Consider Example 3, Competitive Hunter Model Spotted Owls and Hawks. Experiment with different values for the coefficients using the starting values given. Then try different starting values. What is
Continuation of Example 4, Section 1.2: Find the equilibrium value of the digoxin model. What is the significance of the equilibrium value?Data from example 4Digoxin is used in the treatment of heart
``The Relationship between Directional Heading of an Automobile and Steering Wheel Deflection,'' by John E. Prussing, UMAP 506. This unit develops a model relating the compass heading and the
``Radioactive Chains: Parents and Daughters,'' by Brindell Horelick and Sinan Koont, UMAP 234. When a radioactive substance A decays into a substance B, A and B are called parent and daughter. It
``Kinetics of Single Reactant Reactions,'' by Brindell Horelick and Sinan Koont, UMAP 232. The unit discusses reaction orders of irreversible single reactant reactions. The equation a′(t) =
The gross national product (GNP) represents the sum of consumption purchases of goods and services, government purchases of goods and services, and gross private investment (which is the increase in
Consider launching a satellite into orbit using a single-stage rocket. The rocket is continuously losing mass, which is being propelled away from it at significant speeds. We are interested in
a. Using the estimate that db = 0.054v2, where 0.054 has dimension ft hr2/mi2, show that the constant k in Equation (11.29) has the value 19.9 ft/sec2.b. Using the data in Table 4.4, plot db in ft
``Modeling the Nervous System: Reaction Time and the Central Nervous System,'' by Brindell Horelick and SinanKoont,UMAP67. The module models the process by which the central nervous system reacts to
``Tracer Methods in Permeability,'' by Brindell Horelick and Sinan Koont, UMAP 74. This module describes a technique for measuring the permeability of red corpuscle surfaces to K42 ions, using
``Selection in Genetics,'' by Brindell Horelick and Sinan Koont, UMAP 70. This module introduces genetic terminology and basic results about genotype distribution in successive generations. A
Write a summary report on the article ``Case Studies in Cancer and Its Treatment by Radiotherapy,'' by J. R. Usher and D. A. Abercrombie, International Journal of Mathematics Education in Science
A patient is given a dosage Q of a drug at regular intervals of time T . The concentration of the drug in the blood has been shown experimentally to obey the law:a. If the first dose is administered
Sketch how a series of doses might accumulate based on the concentration curve given in Figure 11.14.Figure 11.14
Suggest other phenomena for which the model described in the text might be used.
Suppose that k = 0.2 hr–1 and that the smallest effective concentration is 0:03 mg/ml. A single dose that produces a concentration of 0.1 mg/ml is administered. Approximately how many hours will
Suppose k = 0.01 hr–1and T = 10 hr. Find the smallest n such that Rn > 0.5R.
a. If k = 0.05 hr–1 and the highest safe concentration is e times the lowest effective concentration, find the length of time between repeated doses that will ensure safe but effective
Discuss how the elimination constant k in Equation (11.15) could be obtained experimentally for a given drug.Equation 11.15
Complete the requirements of the UMA P module ``The Cobb Douglas Production Function,'' by Robert Geitz, UMAP 509. A mathematical model relating the output of an economic system to labor and capital
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
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
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.
The following data were obtained for the growth of a sheep population introduced into a new environment on the island of Tasmania (adapted from J. Davidson, ``On the Growth of the Sheep Population in
Consider the solution of Equation (11.8). Evaluate the constant C in Equation (11.10) in the case that P > M for all t. Sketch the solutions in this case. Also sketch a solution curve for the case
Prepare a report on the ``Geometry of the Arms Race,'' UMAP 311.
Research the Cuban missile crisis of 1962. Determine the possible strategies of the United States and the Soviet Union. Provide values to your payoff matrix. Determine what each side should do.
Guerillas plan on attacking a police compound. Assume the size of the guerilla force is m, and the size of the police force is n. Further assume they are using the same weaponry.If the size of the
Golf (give the putt or not) involving a professional golf match is the topic of this problem.In matches, like the President's Cup, the Ryder Cup, and other match play events, it is common to see
Doc Holliday versus Ike Clanton (2-person, 3-strategy game) are used in this problem. On October 26, 1881, the bad blood between the Earps, Clantons, and McLaurys came to a head at the O.K. Corral.
Colin must defend two cities with one indivisible regiment of soldiers. His enemy, Rose, plans to attack one city with her indivisible regiment. City I have a value of 10 units, while City II has a
Consider a ``duel'' between two players. Let's call these players H and D. Now, we have historical information on each because this is not their first duel. H will kill at long range with
Create your own scenario that follows the format of the game of Battle of the Sexes, Chicken, or the Prisoner's Dilemma. Identify all strategies; use ordinal ranking to give them value. Completely
Consider the following 2 × 2 non-zero-sum game:a. Find the solution if both players play their maximin strategies.b. Apply strategic moves to see if either player can improve his outcome.
Use movement diagrams to find all the stable outcomes in Problems 1 through 5. Then use strategic moves (using Table 10.2) to determine if Rose can get a better outcome.Table 10.2Consider the
Use movement diagrams to find all the stable outcomes in Problems 1 through 5. Then use strategic moves (using Table 10.2) to determine if Rose can get a better outcome.Table 10.2
Use movement diagrams to find all the stable outcomes in Problems 1 through 5. Then use strategic moves (using Table 10.2) to determine if Rose can get a better outcome.Table 10.2
Use movement diagrams to find all the stable outcomes in Problems 1 through 5. Then use strategic moves (using Table 10.2) to determine if Rose can get a better outcome.Table 10.2
Use movement diagrams to find all the stable outcomes in Problems 1 through 5. Then use strategic moves (using Table 10.2) to determine if Rose can get a better outcome.Table 10.2
Solve the batter-pitcher duel for the following players:In Problems 11–14, solve the games.
In Problems 11–14, solve the games.
In Problems 11–14, solve the games.
In Problems 11–14, solve the games.
In the game in Problem 9, show the value of the game is:Data from problem 9Given:where a > d > b > c. Show that if Colin plays C1 and C2 with probabilities y and (1 —y) that:
Given:where a > d > b > c. Show that if Colin plays C1 and C2 with probabilities y and (1 —y) that:
Use the alternative methods(a) Equating expected value and (b) Methods of oddments to find the solution to the following games. Assume the row player is maximizing his payoffs which are shown in the
Use the alternative methods (a) equating expected value and (b) methods of oddments to find the solution to the following games. Assume the row player is maximizing his payoffs which are shown in the
Use the alternative methods (a) equating expected value and (b) methods of oddments to find the solution to the following games. Assume the row player is maximizing his payoffs which are shown in the
Use the alternative methods: (a) Equating expected value and (b) Methods of oddments to find the solution to the following games. Assume the row player is maximizing his payoffs which are shown in
Use the alternative methods (a) equating expected value and (b) methods of oddments to find the solution to the following games. Assume the row player is maximizing his payoffs which are shown in the
In the following problems, use the maximin and minimax method and the movement diagram to determine if any pure strategy solutions exist. Assume the row player is maximizing his payoffs which are
In the following problems, use the maximin and minimax method and the movement diagram to determine if any pure strategy solutions exist. Assume the row player is maximizing his payoffs which are
In the following problems, use the maximin and minimax method and the movement diagram to determine if any pure strategy solutions exist. Assume the row player is maximizing his payoffs which are
In the following problems, use the maximin and minimax method and the movement diagram to determine if any pure strategy solutions exist. Assume the row player is maximizing his payoffs which are
We are considering one of three alternatives A, B, or C under uncertain conditions. The payoff matrix is as follows:
Alocal investor is considering three alternative real estate investmentsa hotel, a restaurant, and a convenience storefor a new development area. The hotel and the convenience store will be adversely
We are considering three alternatives A, B, or C or a mix of the three alternatives under uncertain conditions of the economy. The payoff matrix is as follows:
In tennis, statistics are kept for percentages of successful return of first serves by fore-hand and backhand. Choose two tennis players and determine the best strategy for each player.
Take your favorite baseball batter and pitcher from the same league and era. Find batting and pitching statistics for each player. Determine the best strategy for each in a head-to-head competition.
Suppose the payoff matrix of prey survival probabilities is as follows:Analyze this game and present the results. Which predators and prey might fit this scenario? Research two predator-prey species
For problems ag build a linear programming model for each player's decisions and solve it both geometrically and algebraically. Assume the row player is maximizing his payoffs which are show in the

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