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a first course in mathematical modeling

A First Course In Mathematical Modeling 5th Edition Frank R. Giordano, William P. Fox, Steven B. Horton - Solutions

How many possible intersection points are there in the following cases?a. 2 decision variables and 5 ≤ inequalities.b. 2 decision variables and 10 ≤ inequalities.c. 5 decision variables and 12 ≤ inequalities.d. 25 decision variables and 50 ≤ inequalities.e. 2000 decision variables and 5000
Fit the model to the data using Chebyshev's criterion to minimize the largest deviation.a. y = cx
Optimize 5x + 3y subject toFor Problems 8–12, find both the maximum solution and the minimum solution using graphical analysis. Assume x ≥ 0 and y ≥0 for each problem.
Optimize x– y subject toFor Problems 8–12, find both the maximum solution and the minimum solution using graphical analysis. Assume x ≥ 0 and y ≥0 for each problem.
Optimize 6x + 5y subject toFor Problems 8–12, find both the maximum solution and the minimum solution using graphical analysis. Assume x ≥ 0 and y ≥0 for each problem.
Optimize 6x + 4y subject toFor Problems 8–12, find both the maximum solution and the minimum solution using graphical analysis. Assume x ≥ 0 and y ≥0 for each problem.
Optimize 2x + 3y subject toFor Problems 8–12, find both the maximum solution and the minimum solution using graphical analysis. Assume x ≥ 0 and y ≥0 for each problem.
Minimize 5x + 7y subject toSolve Problems 4–7 using graphical analysis.
Maximize 10x + 35y subject toSolve Problems 4–7 using graphical analysis.
Minimize x + y subject toSolve Problems 4–7 using graphical analysis.
Maximize x + y subject toSolve Problems 4–7 using graphical analysis.
A Montana farmer owns 45 acres of land. She is planning to plant each acre with wheat or corn. Each acre of wheat yields $200 in profits, whereas each acre of corn yields $300 in profits. The labor and fertilizer requirements for each are provided here. The farmer has 100 workers and 120 tons of
A local company restores cars and trucks for resale. Each vehicle must be processed in the refinishing/paint shop and the machine/body shop. Each car (on average) contributes $3000 to profit, and each truck contributes (on average) $2000 to profit. The refinishing/paint shop has 2400 work-hours
Consider a company that carves wooden soldiers. The company specializes in two main types: Confederate and Union soldiers. The profit for each is $28 and $30, respectively. It requires 2 units of lumber, 4 hr of carpentry, and 2 hr of finishing to complete a Confederate soldier. It requires 3 units
``Municipal Recycling: Location and Optimality,'' by Jannett Highll and Michael McAsey, UMAP Journal Vol. 15(1), 1994. This article considers optimization in municipal recycling. Read the article and prepare a 10-min classroom presentation.For Projects 1–5, complete the requirements in
``Geometric Programming,'' by Robert E. D. Woolsey, UMAP 737. This unit provides some alternative optimization formulations, including geometric programming. Familiarity with basic differential calculus is required.For Projects 1–5, complete the requirements in the referenced UMAP module
The High Cost of Clean Water: Models for Water Quality Management, by Edward Beltrami, UMAP Expository Monograph. To cope with the severe wastewater disposal problems caused by increases in the nation's population and industrial activity, the U.S. Environmental Protection Agency (EPA) has
``Calculus of Variations with Applications in Mechanics,'' by Carroll O. Wilde, UMAP 468. This module provides a brief introduction to finding functions that yield the maximum or minimum value of certain definite integral forms, with applications in mechanics. Students learn Euler's
``Unconstrained Optimization,'' by Joan R. Hundhausen and Robert A. Walsh, UMAP 522. This unit introduces gradient search procedures with examples and applications. Acquaintance with elementary partial differentiation, chain rules, Taylor series, gradients, and vector dot products is
A farm family owns 100 acres of land and has $25,000 in funds available for investment. Its members can produce a total of 3500 work-hours worth of labor during the winter months (mid-September to mid-May) and 4000 work-hours during the summer. If any of these work-hours are not needed, younger
A truck company has allocated $800,000 for the purchase of new vehicles and is considering three types. Vehicle A has a 10-ton payload capacity and is expected to average 45 mph; it costs $26,000. Vehicle B has a 20-ton payload capacity and is expected to average 40 mph; it costs $36,000. Vehicle C
An electronics rm is producing three lines of products for sale to the government: transistors, micromodules, and circuit assemblies. The firm has four physical processing areas designated as follows: transistor production, circuit printing and assembly, transistor and module quality control, and
A candy store sells three different assortments of mixed nuts, each assortment containing varying amounts of almonds, pecans, cashews, and walnuts. To preserve the store's reputation for quality, certain maximum and minimum percentages of the various nuts are required for each type of assortment,
A manufacturer of an industrial product has to meet the following shipping schedule:The monthly production capacity is 30,000 units and the production cost per unit is $10. Because the company does not warehouse, the service of a storage company is utilized whenever needed. The storage company
A rancher has determined that the minimum weekly nutritional requirements for an average-sized horse include 40 lb of protein, 20 lb of carbohydrates, and 45 lb of roughage. These are obtained from the following sources in varying amounts at the prices indicated:Formulate a mathematical model to
You have just become the manager of a plant producing plastic products. Although the plant operation involves many products and supplies, you are interested in only three of the products:(1). A vinyl–asbestos floor covering, the output of which is measured in boxed lots, each covering a certain
Use linear regression to formulate and analyze Projects 1-5 in Section 2.3.Data from project 1In 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 this in the
For Table 2.7, predict weight as a function of the cube of the height.Table 2.7
Use the basic linear model y = ax + b to t the following data sets. Provide the model, provide the values of SSE, SSR, SST, and R2, and provide a residual plot. For Table 2.7, predict weight as a function of height.Table 2.7
Two alternative designs are submitted for a landing module to enable the transport of astronauts to the surface of Mars. The mission is to land safely on Mars, collect several hundred pounds of samples from the planet's surface, and then return to the shuttle in its orbit around Mars. The
Consider a more advanced stereo system with component reliabilities as displayed in Figure 6.13. Determine the system's reliability. What assumptions are required?Figure 6.13
Consider a personal computer with each item's reliability as shown in Figure 6.12. Determine the reliability of the computer system. What assumptions are required?Figure 6.12
Consider a stereo with CD player, FMAM radio tuner, speakers (dual), and power amplfier (PA) components, as displayed with the reliabilities shown in Figure 6.11. Determine the system's reliability. What assumptions are required in your model?Figure 6.11
Consider the pollution in two adjoining lakes in which the lakes are shown in Figure 6.6 and assume the water flows freely between the two lakes, but the pollutants flow as in the Markov state diagram, Figure 6.7. Let an and bn be the total amounts of pollution in Lake A and Lake B, respectively,
In Example 2, it was assumed that initially the voters were equally divided among the three parties. Try several different starting values. Is equilibrium achieved in each case? If so, what is the final distribution of voters in each case?Data from example 2Presidential voting tendencies are of
In Example 1, assume that all cars were initially in Orlando. Try several different starting values. Is equilibrium achieved in each case? If so, what is the final distribution of cars in each case?Data from example 1Consider a rental car company with branches in Orlando and Tampa. Each rents cars
Consider adding a pizza delivery service as an alternative to the dining halls. Table 6.3 gives the transition percentages based on a student survey. Determine the long-term percentages eating at each place.Table 6.3
Consider a model for the long-term dining behavior of the students at College USA. It is found that 25% of the students who eat at the college's Grease Dining Hall return to eat there again, whereas those who eat at Sweet Dining Hall have a 93% return rate. These are the only two dining halls
In the Los Angeles County School District, substitute teachers are placed in a pool and paid whether they teach or not. It is assumed that if the need for substitutes exceeds the size of the pool, classes can be covered by regular teachers, but at a higher pay rate. Letting x represent the number
Pick a traffic intersection with a traffic light. Collect data on vehicle arrival times and clearing times. Build a Monte Carlo simulation to model traffic flow at this intersection.
Write a computer simulation to implement a baseball game between your two favorite teams (see Problem 6).Data from problem 6Construct a Monte Carlo simulation of a baseball game. Use individual batting statistics to simulate the probability of a single, double, triple, home run, or out. In a more
Write a computer simulation to implement the ship harbor algorithm.
Assume a storage cost of $0.001 per gallon per day and a delivery charge of $500 per delivery. Construct a computer code of the algorithm you constructed in Problem 4 and compare various order points and order quantity strategies.Data from problem 4Problem 3 suggests an alternative inventory
Complete the requirements of UMAP module 340, ``The Poisson Random Process,'' by Carroll O.Wilde. Probability distributions are introduced to obtain practical information on random arrival patterns, interarrival times or gaps between arrivals, waiting line buildup, and service loss rates. The
In the case in which a gasoline station runs out of gas, the customer is simply going to go to another station. In many situations (name a few), however, some customers will place a back order or collect a rain check. If the order is not filled within a time period varying from customer to customer
Problem 3 suggests an alternative inventory strategy. When the inventory reaches a certain level (an order point), an order can be placed for an optimal amount of gasoline. Construct an algorithm that simulates this process and incorporates probabilistic submodels for demand and lag times. How
In many situations, the time T between deliveries and the order quantity Q is not fixed. Instead, an order is placed for a specific amount of gasoline. Depending on how many orders are placed in a given time interval, the time to fill an order varies. You have no reason to believe that the
Most gasoline stations have a storage capacity Qmax that cannot be exceeded. Refine the inventory algorithm to take this consideration into account. Because of the probabilistic nature of the demand submodel at the end of the inventory cycle, there might still be significant amounts of gasoline
Modify the inventory algorithm to keep track of unfilled demands and the total number of days that the gasoline station is without gasoline for at least part of the day.
The demand for steel belted radial tires for resupply on a weekly basis is provided in the following table:Assumptions: Lead time for resupply is between 1 and 3 days. Currently, we have 7 steel belted radial tires in stock and no orders are currently due. We seek to determine the order quantity
Let's Make a DealYou are dressed to kill in your favorite costume and the host picks you out of the audience. You are offered the choice of three wallets. Two wallets contain a single $50 bill, and the third contains a $1000 bill. You choose one of the wallets, 1, 2, or 3. The host, who knows
The Price Is Right–On the popular TV game show The Price Is Right, at the end of each half hour, the three winning contestants face off in the Showcase Showdown. The game consists of spinning a large wheel with 20 spaces on which the pointer can land, numbered from $0.05 to $1.00 in 5-cents
In American roulette, there are 38 spaces on the wheel: 0, 00, and 1–36. Half the spaces numbered 1–36 is red, and half is black. The two spaces 0 and 00 are green. Simulate the playing of 1000 games betting either red or black (which pay even money, 1:1). Bet $1 on each game and keep track of
Construct and perform a Monte Carlo simulation of a horse race. You can be creative and use odds from the newspaper, or simulate the Mathematical Derby with the entries and odds shown in following table.Construct and perform a Monte Carlo simulation of 1000 horse races. Which horse won the most
Craps–Construct and perform a Monte Carlo simulation of the popular casino game of craps. The rules are as follows: There are two basic bets in craps, pass and don't pass. In the pass bet, you wager that the shooter (the person throwing the dice) will win; in the don't pass bet, you wager
In the following data, X represents the diameter of a ponderosa pine measured at breast height, and Y is a measure of volume—number of board feet divided by 10 (see Problem 4, Section 4.2).In Problems 6-12, construct a scatterplot of the given data. Is there a trend in the data?Are any of the
In the following data, X is the Fahrenheit temperature and Y is the number of times a cricket chirps in 1 min (see Problem 3, Section 4.2).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? Construct a divided difference
Table 4.7 and Figure 4.9 present data representing the commercial harvesting of oysters in Chesapeake Bay. Fit a simple, one-term model to the data. How well does the best one-term model you find fit the data? What is the largest error? The average error?Table 4.7Figure 4.9 Table 4.7 Oysters in the
Solve Problems 1-4 with the model V = m(logP) + b. Compare the errors with those computed in Problem 4. Compare the two models. Which is better?Data from problem 4From the data in Table 4.6, calculate the mean (i.e., the average) of the Bornstein errors ∣Vobserved – Vpredicted∣. What do the
For the data sets in Problems 1-4, construct a divided difference table. What conclusions can you make about the data? Would you use a low-order polynomial as an empirical model? If so, what order? X y 0 1 1 4.5 20 90 403 2 3 4 4 5 6 7 1808 8103 36,316
In the following data, X represents the diameter of a ponderosa pine measured at breast height, and Y is a measure of volumenumber of board feet divided by 10. Make a scatterplot of the data. Discuss the appropriateness of using a 13th-degree polynomial that passes through the data points as an
From the data in Table 4.6, calculate the mean (i.e., the average) of the Bornstein errors ∣Vobserved – Vpredicted∣. What do the results suggest about the merit of the model?Table 4.6In 1976, Marc and Helen Bornstein studied the pace of life.2 To see if life becomes more hectic as the size
Using the data, a calculator, and the model you determined for V (Problem 1f), complete Table 4.6.Table 4.6Data from problem 1fFit 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
For the data sets in Problems 1-4, construct a divided difference table. What conclusions can you make about the data? Would you use a low-order polynomial as an empirical model? If so, what order? x y 0 1 7 15 33 2 3 4 5 7 61 99 147 205 273 6
For the data sets in Problems 1-4, construct a divided difference table. What conclusions can you make about the data? Would you use a low-order polynomial as an empirical model? If so, what order? x y 012 23 48 3 4 5 6 7 123 148 173 198 00 73 98
Consider the rising cost of postcards over time. The data and scatterplot are provided in the following:Develop a mathematical model to predict:(a). when the cost might be $0.50 and(b). what the cost might be in 2020.For Projects 2-9, use the software you developed in Project 1 to find the splines
For each of the following data sets, write a system of equations to determine the coefficients of the natural cubic splines passing through the given points. If a computer program is available, solve the system of equations and graph the splines.a. b. c. d. X y 2 2 4 8 7 12
For the data sets in Problems 1-4, construct a divided difference table. What conclusions can you make about the data? Would you use a low-order polynomial as an empirical model? If so, what order? x y 0 1 2 3 4 5 6 7 28 24 56 110 192 308 464
Data for the ponderosa pine:a. y = ax + bb. y = ax2c. y = ax3d. y = ax3 + bx2 + cIn Problems 7–10, fit the data with the models given, using least squares. x 17 19 20 22 23 25 28 31 32 33 37 39 42 19 25 32 51 57 71 113 140 153 187 192 205 250 260 36 y S
Use the data presented in Problem 7b to fit the models W =cg3 and W = kgl2. Interpret these models. Compute appropriate indicators and determine which model is best. Explain.Data from problem 7bIn the following data, g represents the girth of a fish. Fit the model W = klg2 to the data using the
Data for stretch of a spring:a. y = ax.b. y = b + ax.c. y = ax2.In Problems 7–10, fit the data with the models given, using least squares. x(x10) | 5 10 20 30 y(x10-5) 0 19 57 94 40 50 60 70 80 90 100 134 173 216 256 297 343 390
In 1601 the German astronomer Johannes Kepler became director of the Prague Observatory. Kepler had been helping Tycho Brahe in collecting 13 years of observations on the relative motion of the planet Mars. By 1609 Kepler had formulated his first two laws:i. Each planet moves on an ellipse with
In Problems 7–10, fit the data with the models given, using least squares.a. y = b + axb. y = ax2 X 1 2 3 y 1 12 4 5 2 4
a. In the following data, W represents the weight of a fish (bass) and l represents its length. Fit the model W = kl3 to the data using the least-squares criterion.b. In the following data, g represents the girth of a sh. Fit the model W = klg2 to the data using the least-squares criterion.
A general rule for computing a person's weight is as follows: For a female, multiply the height in inches by 3:5 and subtract 108; for a male, multiply the height in inches by 4.0 and subtract 128. If the person is small bone-structured, adjust this computation by subtracting 10%; for a large
Problem 6 in Section 3.1Data from problem 6The following data represent (hypothetical) energy consumption normalized to the year1900. Plot the data. Test the model Q=aebx by plotting the transformed data. Estimate the parameters of the model graphically.For Problems 1-6, find a model using the
The 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 data. Estimate the parameters of the following models.a. P = c1t.b. P = aebt. (days) P (number of observed flies) 7 8 14 41 21 133 28 250 35 280
Suppose the variable x1 can assume any real value. Show that the following substitution using nonnegative variables x2 and x3 permits x1 to assume any real value.Thus, if a computer code allows only nonnegative variables, the substitution allows for solving the linear program in the variables x2
In 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 by observing how fast people walk. Plot the data. What kind of a relationship is suggested? Test the following models by plotting the appropriate
For each of the following data sets, formulate the mathematical model that minimizes the largest deviation between the data and the line y = ax+b. If a computer is available, solve for the estimates of a and b.a. b. c. X y 1.0 3.6 2.3 3.0 3.7 3.2 4.2 5.1 6.1 5.3 7.0 6.8
For the following data, formulate the mathematical model that minimizes the largest deviation between the data and the model y = c1x2 + c2x + c3. If a computer is available, solve for the estimates of c1, c2, and c3. X y 0.1 0.06 0.2 0.12 0.3 0.36 0.4 0.65 0.5 0.95
Derive the equations that minimize the sum of the squared deviations between a set of data points and the quadratic model y = c1x2 + c2x + c3. Use the equations to find estimates of c1, c2, and c3 for the following set of data.Compute D and dmax to bound cmax. Compare the results with your
Write a computer program that uses Equations (3.4) and the appropriate transformed data to estimate the parameters of the following models.a. y = bxnb. y = beaxc. y = a lnx + bd. y = ax2e. y = ax3equation 3.4 ΤΗΝ m «Σx +b 2 x = = i=l i=l 777 Σ xivi i=l ΤΡΙ 777 a Σ xi + mb =
In the following data, x is the diameter of a ponderosa pine in inches measured at breast height and y is a measure of volume—number of board feet divided by 10. Test the model y = axb by plotting the transformed data. If the model seems reasonable, estimate the parameters a and b of the model
Use Equations (3.5) and (3.6) to estimate the coefficients of the line y = ax + b such that the sum of the squared deviations between the line and the following data points is minimized.a.b. c. For each problem, compute D and dmax to bound cmax. Compare the results to your solutions to Problem 2
Write a computer program that computes the deviation from the data points and any model that the user enters. Assuming that the model was fitted using the least-squares criterion, compute D and dmax. Output each data point, the deviation from each data point, D, dmax, and the sum of the squared
The 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). Test the model e = c1S by plotting the data. Estimate c1 graphically. S (×10-3) | 5 e(x105) 0 10 19 20 30 40 50 57 134 60 70 80 90 173 216
Problem 3 in Section 3.1Data from Problem 3 In the following data, x is the diameter of a ponderosa pine in inches measured at breast height and y is a measure of volume number of board feet divided by 10. Test the model y = axb by plotting the transformed data. If the model seems reasonable,
Compete the requirements of the module `Curve Fitting via the Criterion of Least Squares,'' by John W. Alexander, Jr., UMAP 321. (See enclosed CD for UMAP module.) This unit provides an easy introduction to correlations, scatter diagrams (polynomial, logarithmic, and exponential scatters), and
Using elementary calculus, show that the minimum and maximum points for y = f (x) occur among the minimum and maximum points for y = f2(x). Assuming f (x) ≥ 0, why can we minimize f (x) by minimizing f 2(x)
Solve the two equations given by (3.4) to obtain the values of the parameters given by Equations (3.5) and (3.6), respectively.Equation 3.4Equation 3.5equation 3.6 MI M m a Σx +6 Σ * | x = 2 xy = 2 Σ i=l i=l 777 HTT a Σ xi + mb = Σ I=! l=!
Write a computer program that finds the least-squares estimates of the coefficients in the following models.a. y = ax2 + bx + cb. y = axn
The model in Figure 3.2 would normally be used to predict behavior between x1 and x5. What would be the danger of using the model to predict y for values of x less than x1 or greater than x5? Suppose we are modeling the trajectory of a thrown baseball.Figure 3.2 5x Tx Ex Zx lx A
Discuss the differences between using a model to predict versus using one to explain a real-world system. Think of some situations in which you would like to explain a system. Likewise, imagine other situations in which you would want to predict a system.For the scenarios presented in Problems
Consider a new company that is just getting started in producing a single product in a competitive market situation. Discuss some of the short-term and long-term goals the company might have as it enters into business. How do these goals affect employee job assignments? Would the company
How should we save a portion of our earnings?For the scenarios presented in Problems 9-17, identify a problem worth studying and list the variables that affect the behavior you have identified. Which variables would be neglected completely? Which might be considered as constants initially? Can you
For the submodel concerning braking distance in the vehicular stopping distance model, how would you design a brake system so that the maximum deceleration is constant for all vehicles regardless of their mass? Consider the surface area of the brake pads and the capacity of the hydraulic system to
How can we improve our ability to sign up for the best classes each term?For the scenarios presented in Problems 9-17, identify a problem worth studying and list the variables that affect the behavior you have identified. Which variables would be neglected completely? Which might be considered as
For the vehicular stopping distance model, design a test to determine the average response time. Design a test to determine average reaction distance. Discuss the difference between the two statistics. If you were going to use the results of these tests to predict the total stopping distance, would
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.

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