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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

y′ + 2xy = xIn Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
y′/2 + y = e–x sin xIn Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
y′=2y/x+ x3ex – 1In Problems 1–15, find the general solution of the given first-order linear differential equation. State an interval over which the general solution is valid.
y dx + (3x –xy + 2) dy = 0, y (2) =–1, y < 0In Problems 16–20, solve the initial value problem.
Oxygen flows through one tube into a liter flask filled with air, and the mixture of oxygen and air (considered well stirred) escapes through another tube. Assuming that air contains 21% oxygen, what percentage of oxygen will the ask contain after 5 L have passed through the intake tube?
If the average person breathes 20 times per minute, exhaling each time 100 in3 of air containing 4% carbon dioxide. Find the percentage of carbon dioxide in the air of a 10,000-ft3 closed room 1 hr after a class of 30 students enters. Assume that the air is fresh at the start, that the ventilators
a. From the Fundamental Theorem of Calculus, we haveUse this fact to show that any function y given by Equation (11.61) solves linear first-order Equation (11.55).Equation 11.61Equation 11.55b. If the constant C is given by:in Equation (11.61), show that the resulting function y defined by Equation
In Problems 1–4, verify that the given function pair is a solution to the first-order system. x = -e, y = el dx dy dt dt =-y. = = -x
In Problems 1–4, verify that the given function pair is a solution to the first-order system. x = e, y =e
In Problems 1–4, verify that the given function pair is a solution to the first-order system. x = dx dt b tanh bt, y =b sech bt, b = any real number dy = = y, = xy dt
In Problems 5–8, find and classify the rest points of the given autonomous system. dx dt = 2y. dy dt =-3x
In Problems 5–8, find and classify the rest points of the given autonomous system. dx dt = -(y-1), dy dt =x-2
In Problems 5–8, find and classify the rest points of the given autonomous system. dx dt =-y(y-1), dy dt = (x - 1)(y-1)
In Problems 5–8, find and classify the rest points of the given autonomous system. ||
Complete the requirements of the UMAP module ``Whales and Krill: A Mathematical Model,'' by Raymond N. Greenwell, UMAP 610. This module models a predatorprey system involving whales and krill by a system of differential equations. Although the equations are not solvable, information is extracted
List three important considerations that are ignored in the development of the competitive hunter model presented in this section.
Consider the competitive hunter model defined by:a. What assumptions are implicitly being made about the growth of trout and bass in the absence of competition?b. Interpret the constants a, b, m, n, k1, and k2 in terms of the physical problem.c. Perform a graphical analysis and answer the
Show that the two trajectories leading to (m, a/b) shown in Figure 12.8 are unique.Figure 12.8a. From system (12.6) derives the following equation:b. Separate variables, integrate, and exponentiate to obtain:c. Let f (y) = y a/ e by and g(x) = x m/e nx. Show that f (y) has a unique maximum ofMy =
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 situation, a natural Australian predator, a ladybird beetle (Novius cardinalis) was imported. The beetles
Consider two species whose survival depends on their mutual cooperation. Let's take as an example a species of bee that feeds primarily on the nectar of one plant species and simultaneously pollinates that plant. One simple model of this mutualism is given by the autonomous system:a. What
Verify that the Function (12.23) satisfies the differential Equation (12.22).Equation 12.22a. Solve Equation (12.29) for y and substitute the results into the differential equation dx/dt = –gxy from Model (12.27).Equation 12.29 b. Separate the variables in the differential equation resulting
Let denote a guerilla force and Y denote a conventional force. The autonomous system:is a Lanchestrian model for conventionalguerilla combat in which there are no operational loss rates and no reinforcements.a. Discuss the assumptions and relationships necessary to justify the model. Does the
a. Assuming that the single-weapon attrition rates a and b in Equations (12.26) are constant over time, discuss the submodels:Equation 12.26where ry and rx are the respective firing rates (shots/combatant/day) of the Y and the X forces, and py and px are the respective probabilities that a single
In our Model (12.36) for the arms race, assume that an– bm < 0, so the rest point lies in a quadrant other than the first one in the phase plane. Sketch the lines dx/dt = 0 and dy/dt = 0 in the phase plane, and label them and their intercepts on the coordinate axes. Perform a graphical
Find the first three approximations (x1, y1), (x2, y2), (x3, y3) using Euler's method for the predator–prey system of whales y and krill x. dx dt = 3x - xy dy dt subject to the initial conditions xo = 1 and yo = 2 starting at to = 0, with At = 0.1. = xy - 2y
In Problems 1–4, use Euler's method to solve the first-order system subject to the specified initial condition. Use the given step size Δt and calculate the first three approximations (x1, y1), (x2, y2), and (x3, y3). Then repeat your calculations for Δt/2. Compare your approximations with the
In Problems 1–4, use Euler's method to solve the first-order system subject to the specified initial condition. Use the given step size Δt and calculate the first three approximations (x1, y1), (x2, y2), and (x3, y3). Then repeat your calculations for Δt/2. Compare your approximations with the
In Problems 1–4, use Euler's method to solve the first-order system subject to the specified initial condition. Use the given step size Δt and calculate the first three approximations (x1, y1), (x2, y2), and (x3, y3). Then repeat your calculations for Δt/2. Compare your approximations with the
In Problems 1–4, use Euler's method to solve the first-order system subject to the specified initial condition. Use the given step size Δt and calculate the first three approximations (x1, y1), (x2, y2), and (x3, y3). Then repeat your calculations for Δt/2. Compare your approximations with the
Assume we have a lake that is stocked with both bass and trout. Because both eat the same food sources, they are competing for survival. Let B(t) and T (t) denote the bass and trout populations, respectively, at time t. The rates of growth for bass and for trout are estimated by the differential
Repeat Problem 5 for step sizeΔt = 1. Discuss the differences in your plots and explain why these differences occur.Data from problem 5Assume we have a lake that is stocked with both bass and trout. Because both eat the same food sources, they are competing for survival. Let B(t) and T (t)
The following system is a predator–prey model in which harvesting occurs for both species. Use Euler's method with step size Δt = 1 over 0 ≤t ≤4 to numerically solve: dx dt dy dt subject to x (0) = and y(0) = 1. 314 = x-xy-4 3 =xy-y - 4x
A modern flu is spread throughout a small community with a fixed population of size n. The disease is spread through contact between infected persons and persons who are susceptible to the disease. Assume that everyone is susceptible to the disease initially and that the community is contained so
Consider an industrial situation in which it is necessary to set up an assembly line.Suppose that each time the line is set up a cost c is incurred. Assume c is in addition to the cost of producing any item and is independent of the amount produced. Suggest submodels for the production rate. Now
Consider a company that allows back ordering. That is, the company notifies customers that a temporary stock-out exists and that their order will be filled shortly. What conditions might argue for such a policy? What effect does such a policy have on storage costs? Should costs be assigned to
Consider an athlete competing in the shot put. What factors influence the length of his or her throw? Construct a model that predicts the distance thrown as a function of the initial velocity and angle of release. What is the optimal angle of release? If the athlete cannot maximize the initial
John Smith is responsible for periodically buying new trucks to replace older trucks in his company's fleet of vehicles. He is expected to determine the time a truck should be retained so as to minimize the average cost of owning the truck. Assume the purchase price of a new truck is $9000 with
A cow currently weighs 800 lb and is gaining 35 lb per week. It costs $6.50 a week to maintain the cow. The market price today is $0.95 per pound but is falling $0.01 per day. Formulate a mathematical model and find the optimal period to keep the cow until it is sold to maximize profits.
``The Human Cough,'' by Philip M. Tuchinsky, UMAP 211. A model is developed showing how our bodies contract the windpipe during a cough to maximize the velocity of the airow (making the cough maximally effective). Complete the module and prepare a short report for classroom discussion.
``Five Applications of MaxMin Theory from Calculus,'' by Thurmon Whitley, UMAP 341. In this module several unconstrained optimization problems are solved using the calculus. Scenarios addressed include maximizing profit, minimizing cost, minimizing travel time of light as it passes through several
Find the local maximum value of the function: f(x, y) = xy - x - y - 2x - 2y +4
Find the local minimum value of the function: f(x, y) = 3x + 6xy + 7y - 2x + 4y
A differentiable function f (x, y) has a saddle point at a point (a, b) where its partial derivatives are simultaneously zero, if in every open disk centered at (a,b) there are domain points where f (x, y) > f (a, b) and domain points where f (x, y) a. b. f(x, y) = x-y-2xy +6
A continuous function f (x, y) takes on its absolute extrema on a closed and bounded region either at an interior point or at a boundary point of the region. Find the absolute extrema of: f(x, y) = 48xy - 32x - 24y on the square region 0x 1 and 0 y 1.
A company manufactures x floor lamps and y table lamps each day. The profit in dollars for the manufacture and sale of these lamps is: P(x, y) = 18x +2y-0.05x -0.03y +0.02xy - 100 Find the daily production level of each lamp to maximize the company's profits.
If x and y are the amounts of labor and capital, respectively, to produce: Q(x, y) = 0.54x -0.02x+1.89y2-0.09y units of output for manufacturing a product, find the values of x and y to maximize Q.
The total cost to manufacture one unit of product A is $3, and for one unit of product B it is $2. If x and y are the retail prices per unit of A and B, respectively, then marketing research has established that:are the quantities of each product that will be sold each day. Find a function P (x, y)
An electric power-generating company charges different rates for residential and business users. (You might consider some reasons why this would be so.) The cost of producing the electricity is the same for all users and equals $1000 in fixed costs plus an additional $200 for each unit produced. If
Using the data provided in Example 2, fit the nonlinear model y = ae–bx. How does this solution compare to the power model found in Example 2?Data from example 2Consider the following data on the average tread on a radial racing tire over time. The variable x will be the hours in heavy racing
Write a computer code to perform the gradient method of steepest ascent algorithm using the multiplier technique λk+1 =₰ λk discussed in this section. Use your code to solve Problems 1, 5, 6, and 7 of this section.Data from problem 1Find the local maximum value of the function:Data from problem
Resolve the oil transfer problem when the storage capacity is 25 ft3. How does this result compare with our estimated value?
Resolve the water container problem when the surface area available is 500 ft2 and the radius is 9 ft.
Find the hottest point (x, y, z) along the elliptical orbit:Use the method of Lagrange multipliers to solve Problems 3–6. 4x + y + 4z = 16 where the temperature function is T(x, y, z) = 8x + 4yz - 16z + 600
A Least Squares Plane–Given the four points (xk,yk,zk)Use the method of Lagrange multipliers to solve Problems 3–6. (0, 0, 0), (0, 1, 1), (1, 1, 1), (1, 0, -1) find the values of A, B, and C to minimize the sum of squared errors k=1 (Axx+Byk +C - zk) if the points must lie in the plane z =
Resolve the oil transfer problem if we introduce a second storage tank that can be filled to its capacity of 30 ft3. (You might reformulate with four variables xij , the amount of oil type i stored in storage tank j.)
``Lagrange Multipliers and the Design of Multistage Rockets,'' by Anthony L. Peressini, UMAP 517. The method of Lagrange multipliers is applied to compute the minimum total mass of an n-stage rocket capable of placing a given payload in an orbit at a given altitude above the earth's surface.
``Lagrange Multipliers: Applications to Economics,'' by Christopher H. Nevison,UMAP 270. The Lagrange multipliers method is interpreted and studied as the marginal rate of change of a utility function. Differential calculus through Lagrange multipliers is required.
Assume that the environmental carrying capacity Nu is determined principally by the availability of food. Argue that under such an assumption, as N approaches Nu the physical condition of the average fish deteriorates as competition for the food supply becomes more severe. What does this suggest
In 1981 and 1982, the deer population in the Florida Everglades was very high. Although the deer were plentiful, they were on the brink of starvation. Hunting permits were issued to thin out the herd. This action caused much furor on the part of environmentalists and conservationists. Explain the
Suppose Nu < NL. What does this inequality suggest about the economic feasibility of fishing that species? Give several examples.
Figure 13.16 suggests that market forces tend to drive the population to NL. Use that figure to show how taxation or subsidization may be used to control the location of NL.What forms might the taxation and subsidization take? (One cost to the fisherman is the various taxes he pays.) Apply your
A constant price has been assumed in all the models developed in this section. Suggest some fisheries for which that assumption is not realistic. How might you alter the assumption? How would you determine the appropriate tax?
Determine whether the equation:is dimensionally compatible, if s is the position (measured vertically from a fixed reference point) of a body at time t, s0 is the position at t = 0, v0 is the initial velocity, and g is the acceleration caused by gravity. S = So + vot-0.5gt
For the damped pendulum:a. Assume that F is proportional to v2 and use dimensional analysis to show that t= √r/gh(θ, rk1/m).b. Assume that F is proportional to v2 and describe an experiment to test the model t= √r/gh(θ, rk1/m).
a. Use dimensional analysis to establish the cube-root law:for scaling of explosions, where r is the radius or depth of the craterρ, is the density of the soil medium, and W the mass of the explosive.b. Use dimensional analysis to establish the one-fourth-root law:for scaling of explosions, where
Complete the requirements for the module, ``Listening to the Earth: Controlled Source Seismology,'' by Richard G. Montgomery, UMAP 292-293. This module develops the elementary theory of wave reflection and refraction and applies it to a model of the earths subsurface. The model shows how
A model of an airplane wing is tested in a wind tunnel. The model wing has an 18-in. chord, and the prototype has a 4-ft chord moving at 250 mph. Assuming the air in the wind tunnel is at atmospheric pressure, at what velocity should wind tunnel tests be conducted so that the Reynolds number of the
Consider the pace of life'' data from Problem 1, Section 4.1. Consider fitting a 14th- order polynomial to the data. Discuss the disadvantages of using the polynomial to make predictions. If a computer is available, determine and graph the polynomial.Data from problem 1 in section 4.1Fit
Compete the requirements of UMAP 551, ``The Pace of Life, An Introduction to Model Fitting,'' by Bruce King. Prepare a short summary for classroom discussion.
Construct a direction field and sketch a solution curve for the following differential equations:a. b. c. d. e. f. dy/dx = y
Given H = 2 mg/ml, L= 0.5 mg/ml, and k = 0.02 hr–1, suppose concentrations below L are not only ineffective but also harmful. Determine a scheme for administering this drug (in terms of concentration and times of dosage).
The modern philosopher Jean-Jacques Rousseau formulated a simple model of population growth for eighteenth-century England based on the following assumptions:The birthrate in London is less than that in rural England. The death rate in London is greater than that in rural England. As England
a. Show that the population P in the logistic equation reaches half the maximum population M at time t∗ given by:b. Derive the form given by Equation (11.12) for population growth according to the logistic law.Equation 11.12c. Derive the equation ln[P/(M–P)] = r M t–r M t∗ from Equation
Consider the Writers Guild strike example. Let the Writers Guild reconsider its position after 6 months, and the payoffs become as shown in the following payoff matrix.Determine the best strategies to play. Writers Guild/Management payoff matrix Writers Guild (Rose) S NS Management (Colin) IN SQ
Consider the Battle of the Bismarck Sea and assume the intelligence estimates for the number of days available were incorrect due to possible bad weather in the region. If the updated estimates were as follows, what should each side do? North South Imamura North 2.5, -2.5 South 2.75,-2.75 Kenney
Given the payoff matrix below from the Battle of the Sexes: Rose RI R2 C1 (4,3) (1, 1) Colin C2 (2, 2) (3,4)
What assumptions have to be true for there not to be a saddle point solution? Show that the two largest entries must be diagonally opposite each other. Payoffs Rose R1 R2 C1 a b Colin C2 C d
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 matrices below. Rose R1 R2 Colin C1 17.3 -4.6 C2 11.5 20.1
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 matrices below. Rose RI R2 C1 4 -2 Colin C2 -4 -1
Find all pure strategy solutions to the following games:a.b.c. Rose R1 R2 R3 C1 320 Colin C2 1 1 0 C3 4 3 0 C4 1 0 0
Find all pure strategy solutions to the following game: Rose R1 R2 R3 R4 CI 4 -8 7 0 C2 325800 8 Colin C3 1 1 -3 C4 5 1 3 -6
A predator has two strategies for catching a prey (ambush or pursuit). The prey has two strategies for escaping (hide or run). The payoff matrix entries, the chance of survivability, allow the prey to maximize and the predator to minimize. Rose R1 R2 C1 1 5 Colin C2 3 2
A predator has two strategies for catching a prey (ambush or pursuit). The prey has two strategies for escaping (hide or run). The payoff matrix entries, the chance of survivability, allow the prey to maximize and the predator to minimize. Rose R1 R2 C1 5 3 Colin C2 1 0
A predator has two strategies for catching a prey (ambush or pursuit). The prey has two strategies for escaping (hide or run). The payoff matrix entries, the chance of survivability, allow the prey to maximize and the predator to minimize. Prey Run Hide Predator Ambush 0.20 0.80 Pursue 0.40 0.60
Golf Smart sells a particular brand of drivers for $200 each. During the next year, it estimates that it will sell 15, 25, 35, or 45 drivers with respective probabilities of 0.35, 0.25, 0.20, and 0.20. It can buy drivers only in lots of 10 from the manufacturer. Batches of 10, 20, 30, 40, and 50
We have a choice of two investment strategies, stocks and bonds. The returns for each under two possible economic conditions are as follows: Alternative Stocks Bonds Condition 1 $10,000 -$7000 Condition 2 -$4000 -$2000 What decision would you choose? Explain your rationale using game theory.
Given the following payoff matrix for building at possible sites A, B, and C under varying environmental conditions {#1, #2, #3}, set up and then solve the games for the builder and environmental conditions: Alternative Build at A Build at B Build at C #1 $1000 $800 $700 Conditions #2 $2000 $1200
Given the following payoff matrix for alternatives A, B, and C under states of nature #1, #2, #3, and #4, set up and solve both the investor's and nature's game: States of Nature Investor's choices Alternatives A B C Condition #1 1100 850 700 Condition #2 900 1500 1200 Condition #3 400 1000 500
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 shown in the matrices below.a. b. c. d. e. A predator has two strategies for catching a prey (ambush or pursuit).
Use the differential equation model formulated in the preceding problem to answer the following:a. From the derivative evaluated at t = 0, determine an equation of the tangent line T passing through the point (0, 100).b. Estimate Q (1) by finding T (1), where Q(t) denotes the amount of money in
``Feldman's Model,'' by Brindell Horelick and Sinan Koont, UMAP 75. This unit develops a version of G. A. Feldman's model of growth in a planned economy in which all the means of production are owned by the state. Originally, the model was developed by Feldman in connection with planning the
``The Digestive Process of Sheep,'' by Brindell Horelick and Sinan Koont, UMAP 69. This unit introduces a differential equation model for the digestive processes of sheep.The model is tested and fit using collected data and the least-squares criterion.Complete the requirements of the indicated UMAP
Consider the improved Euler's method that averages two slopes, the slope obtained at the beginning of the step and the slope obtained at the end of the step, to improve our accuracy. Assume yn+1=yn+(h/2) ∗[g(tn, yn)+g(tn+1, yn+1)]. Let y′=0.25ty, y (0) = 2, and use the improved Euler's method
Solve y′ = 3x2e–y.
Solve y′ = 2 (x + y2x).
Solve sec x dy –x cot y dx = 0.
Solve e–xy′ = x.
Solve e x+y y′ = x.
Solve the differential equation dy/dx = ln x, where x > 0.
Solve the initial value problem x2yy′ = e y, where y (2) = 0.

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