- •Preface
- •1.1 Introduction
- •1.2 Models and modelling
- •1.3 The learning process for mathematical modelling
- •Summary
- •Aims and objectives
- •2.1 Introduction
- •2.2 Examples
- •2.3 Further examples
- •Appendix 1
- •Appendix 2
- •Aims and objectives
- •3.1 Introduction
- •3.2 Definitions and terminology
- •3.3 Methodology and modelling flow chart
- •3.4 The methodology in practice
- •Background to the problem
- •Summary
- •Aims and objectives
- •4.1 Introduction
- •4.2 Listing factors
- •4.3 Making assumptions
- •4.4 Types of behaviour
- •4.5 Translating into mathematics
- •4.6 Choosing mathematical functions
- •Case 1
- •Case 2
- •Case 3
- •4.7 Relative sizes of terms
- •4.8 Units
- •4.9 Dimensions
- •4.10 Dimensional analysis
- •Summary
- •Aims and objectives
- •5.1 Introduction
- •5.2 First-order linear difference equations
- •5.3 Tending to a limit
- •5.4 More than one variable
- •5.5 Matrix models
- •5.6 Non-linear models and chaos
- •5.7 Using spreadsheets
- •Aims and objectives
- •6.1 Introduction
- •6.2 First order, one variable
- •6.3 Second order, one variable
- •6.4 Second order, two variables (uncoupled)
- •6.5 Simultaneous coupled differential equations
- •Summary
- •Aims and objectives
- •7.1 Introduction
- •7.2 Modelling random variables
- •7.3 Generating random numbers
- •7.4 Simulations
- •7.5 Using simulation models
- •7.6 Packages and simulation languages
- •Summary
- •Aims and objectives
- •8.1 Introduction
- •8.2 Data collection
- •8.3 Empirical models
- •8.4 Estimating parameters
- •8.5 Errors and accuracy
- •8.6 Testing models
- •Summary
- •Aims and objectives
- •9.1 Introduction
- •9.2 Driving speeds
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Rewritten problem statement
- •Obtain the mathematical solution
- •9.3 Tax on cigarette smoking
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •9.4 Shopping trips
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •Interpret the mathematical solution
- •Using the model
- •9.5 Disk pressing
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •Interpret the mathematical solution
- •Further thoughts
- •9.6 Gutter
- •Context and problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •9.7 Turf
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •Interpret the solution
- •9.8 Parachute jump
- •Context and problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •9.9 On the buses
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •9.10 Further battles
- •Discrete deterministic model
- •Discrete stochastic model
- •Comparing the models
- •9.11 Snooker
- •Context
- •Problem statement
- •Formulate a mathematical model
- •Obtain the mathematical solution
- •Interpret the mathematical solution
- •9.12 Further models
- •Mileage
- •Heads or tails
- •Picture hanging
- •Motorway
- •Vehicle-merging delay at a junction
- •Family names
- •Estimating animal populations
- •Simulation of population growth
- •Needle crystals
- •Car parking
- •Overhead projector
- •Sheep farming
- •Aims and objectives
- •10.1 Introduction
- •10.2 Report writing
- •Preliminary
- •Main body
- •Appendices
- •Summary
- •General remarks
- •10.3 A specimen report
- •Contents
- •1 PRELIMINARY SECTIONS
- •1.1 Summary and conclusions
- •1.2 Glossary
- •2 MAIN SECTIONS
- •2.1 Problem statement
- •2.2 Assumptions
- •2.3 Individual testing
- •2.4 Single-stage procedure
- •2.5 Two-stage procedure
- •2.6 Results
- •2.7 Regular section procedures
- •2.8 Conclusions
- •3 APPENDICES
- •3.1 Possible extensions
- •3.2 Mathematical analysis
- •10.4 Presentation
- •Preparation
- •Giving the presentation
- •Bibliography
- •Solutions to Exercises
- •Chapter 2
- •Example 2.2 – Double wiper overlap problem
- •Chapter 4
- •Chapter 5
- •Chapter 6
- •Chapter 8
- •Index
Chapter 4
Temperature outside.
4.1Cost.
Heat saving
4.2See chapter 3. Height of thrower.
4.3Angle of projection. Speed of throw.
4.4Number of lifts available.
Correct lift positions, number of other users, speed of operation of lifts.
(a)(i) No. (ii) Yes. (iii) No. (iv) Yes.
(b)(i) Yes. (ii) No. (iii) No. (iv) No.
(c)(i) No. (ii) No. (iii) Yes. (iv) No.
(d)(i) No. (ii) No. (iii) Yes. (iv) No.
4.5
(e)(i) No. (ii) No. (iii) Yes. (iv) No.
(f)(i) Yes. (ii) No. (iii) No. (iv) No.
(g)(i) No. (ii) Yes. (iii) No. (iv) Yes.
(h)(i) No. (ii) Yes. (iii) No. (iv) No.
(a)(i) Yes. (ii) No. (iii) Yes.
4.6(b) (i) No. (ii) Yes. (iii) No.
(c)(i) Yes. (ii) Yes. (iii) Yes.
(i) a gives a vertical shift.
b affects magnitude of expression.
c affects rate of decay.
4.7
(ii) a magnitudes the expression. b shifts maximum point.
c gives a vertical shift.
4.8(c) is correct.
4.9(e) is correct.
4.10kLU 2 / D.
[ k ] = ML −1 . kg m −1 .
4.11A + B sin(π t /12) + c sin(π t /4380).
4.12A ( x ) = a/x, B ( y ) = b/y where a/x + b/y = c.
4.13A sin 2 (π t /6) exp(−0.1 t ).
(i)c is the smallest term, and the largest term.
4.14(ii) ab is the smallest term, and the largest term.
(iii)c is the smallest term, and the largest term.
4.15(i) x 2 + a. (ii) (iii)
4.16(i) 0.01 x 3 . (ii) (iii)
4.17
(i) 0.001 + 0.0001 x. (ii) x. (iii) 4.18 1.79 m s −1
4.19 1 N = 10 5 dyn = 7.23 pdl.
4.20 1 million ft 3 day −1 = 0.3278 m 3 s −1 .
4.21 The track with a perimeter of 440 yd is longer by 2.34 m. 4.22 7.27 × 10 −5 rad s −1 , 6.67 × 10 4 mph.
4.23 Nm 2 kg −2
4.24 ML −7 T −2
4.25(a) Correct (b) Error (c) Correct (d) Error
4.26(a) Error (b) Correct (c) Error
4.27Incorrect
4.28 and c = 0. Thus u =
Chapter 5
5.2F n +1 = 1.15 F n − 2000 − kP n , P n +1 = 0.99997 P n
5.3A, E, G
5.4E, G
5.5(a) X n +1 = (1 + r /100) X n − 12 m
5.6A n +1 = A n + 15, B n +1 = B n − 9, P n +1 = P n + 5 − 790 P n / A n , Q n +1 = Q n +1 + 790 P n / A n − 800 Q n / B n
(a)X n +1 = (19 X n + 100)/21
5.7(b) X n = 50[1 − (19/21) n ], Y n = 100 − X n
(c)X n and Y n become 50
Chapter 6
6.1 t measured in years, N in thousands.
6.2T ( t ) = 18 + 42 exp(−0.09060 t ). Cools to 30°C in 13.82 min. T (10) = 34.97°C.
6.3V ( t ) = (1.279 − 10.0399 t ) 3 . V is in m 3 , t is in h.
6.4Y = α 2 /4β.
(a)False. (b) False. (c) False. (d) False.
6.5(e) (i) No. (ii) Velocity zero. (iii) No.
(f)(i) No. (ii) Yes. (iii) Yes. (iv) No.
6.6(a) Moment of applied force about hinge axis is required. This is the product of force and perpendicular distance. So, if the distance is reduced then the force must be increased for the same
effect.
(b)Friction opposes motion and is proportional to the normal reaction between the object and the ground. This reaction is larger than pushing.
(c)Circular motion has force towards the centre which increases as the square of the angular velocity.
(d)As capsule moves in an approximately circular orbit, the astronaut experiences a balance between gravitation and centrifugal force.
The differential equation has the form
6.7
where x is the vertical car body displacement, y is the ramp profile, λ is the spring constant, k is the damper constant, n is the speed of the car at time t after the ramp is hit and m is the mass of the car.
For case A, the motion can be described by the equations
and
6.8
where T is the force between the arm and the door, G is the spring force, r = PB, b = BS, l = PS, θ is the angle at which the door is open, π = BSP and I is the moment of inertia of the door. [case B – no solution supplied]
Variable mass problem. The equations of motion are
6.9
and
where x is the length of vertical chain, l the total chain length, T the tension in the chain at the deck edge, ρ the mass per unit length of the chain, μ the coefficient of friction and v the velocity.