Modelling-Module 1 Lecture 2

1. Modelling-Module 1Lecture 2 David Godfrey

2. Modelling Skills • Administration details • Continuing to build the model…the next four steps • Checking relative importance of components • Classifying variables • Checking by approximating • Modelling examples

3. The Next Four Steps • Solve the mathematical equations • Does the solution make mathematical sense? • Does the “correct solution” match reality? • Write a report

4. Write a Report Prepare a report describing the problem and its outcomes. Initial Considerations: • How much detail is required in the report? • How can report be constructed so that important features are clear and the main results stand out? • Who is the report for and what do the readers want to know?

5. Structure of the Report • Title page • Acknowledgements • Contents • Summary • Glossary • Problem statement • …...

6. Structure of the Report (cont). • Assumptions • Data • Model development and solutions • Results • Conclusions and/or recommendations • Appendices • References and/or bibliography

7. Relative Importance of Components • Orders of magnitude…drop small terms? • Thickness of material vs capacity of container vs volume of material vs cost of material

8. Classifying Variables • We want to construct a model which will enable us to work out how far a golf ball travels when hit by a driving club. • Such a model should help us improve the design of the club

9. Gravity Speed of ball off club Size of ball Colour of ball Angle at which ball is hit Wind speed Length of grass Mass of club head Material of ball Wind direction Name of golfer’s dog Slope of fairway Distance ball goes Air density Speed of club head Spin of ball Mass of Ball List “any” Variables

10. Gravity Speed of ball off club Size of ball Colour of ball Angle at which ball is hit Wind speed Length of grass Mass of club head Material of ball Wind direction Name of golfer’s dog Slope of fairway Distance ball goes Air density Speed of club head Spin of ball Mass of Ball What’s Importanti -input, o-output, * irrelevant i o i * o i i i i i * i o i i o i

11. Approximations • Consider the following problem

12. What can the expression be replaced by, experimenting with various magnitudes of and e.g. • If both are very small • If is very small and is very large • If both are very large • The course notes capture the main ideas

13. Example 1

14. Example 1 Ignore squared terms and above – consider effect on each line of fraction

15. Example 1

16. Example 2

17. Select only the highest power terms in each line of the fraction Example 2

18. Example 2

19. Example 3

20. y big x small Example 3

21. y big x small Example 3

22. y = kxz Proportionality • in mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. • We can also say y is proportional to x,z and write y  x,z when there is a straight line relationship between y and xz i.e. y = kxz Y y  xz k is called the constant of proportionality It corresponds to the gradient of the line XZ

23. Proportion example 1 In an industrial experiment it is found that yield y kg is proportional to the temperature t °C, and the mass m, in mg of a catalyst. If a temperature of 18 °C and a mass of 15mg of catalyst gave a yield of 1.84 kg, what would the expected yield be at 20 °C if the mass of catalyst was reduced to 1mg.

24. Proportion example 1 Write down proportionality relations Use given information to calculate k Answer question

25. 1 µ y x 1 = y k x Inverse proportion • We can also talk about inversely proportional to • We say y is inversely proportional to x and write y  x when there is a straight line relationship between y and 1/x

26. Proportion example 2 The energy efficiency E,of a machine is proportional to the number of maintenance checkups per year n, and is inversely proportional to the mass of material produced m tonnes, and the age of the machine a years. If a 5 year old machine producing 20 tonnes of material per year which is checked each month has a efficiency of 75%, give a general expression for efficiency.

27. Proportion example 2 Write down proportionality relations Use given information to calculate k Answer question

28. Modelling example 1 The brightness, b, of a planet as seen from earth varies: • directly as the albedo, a, directly as the phase, p • directly as the square of the planet’s distance from the sun • inversely as the square of the planet’s distance from the sun, s, • and inversely as the square of the planet’s distance from the earth, e. (the albedo is the proportion of sunlight reflected from the surface and the phase is the proportion of the planet’s disc illuminated by the sun). Write down a model for b in terms of the other variables, incorporating all of the above information.

29. Modelling example1 The brightness, b, of a planet as seen from earth varies: • directly as the albedo, a, directly as the phase, p • directly as the square of the planet’s distance from the sun • inversely as the square of the planet’s distance from the sun, s • and inversely as the square of the planet’s distance from the earth, e.

30. Modelling example 2 A newly created volcanic island is gradually colonised by species arriving from the mainland. Let Ni represent the number of species on the island i years after the eruption. Assume that the number of new species arriving each year is: • proportional to the area, A, of the island; • proportional to the difference between the number of species on the island and the number of species, Ns, on the mainland; • inversely proportional to the number of species already on the island; • inversely proportional to the square of the distance, d, from the island to the mainland. Using these assumptions write down an equation expressing Ni+1 in terms of Ni.

31. Modelling example 2 The number of new species arriving each year is: • proportional to the area, A, of the island; • proportional to the difference between the number of species on the island and the number of species, Ns, on the mainland; • inversely proportional to the number of species already on the island; • inversely proportional to the square of the distance, d, from the island to the mainland.

32. Modelling example 2 Now writing in terms of Now writing in terms of

33. Modelling example 3 A ferry has a total deck space of area A. It takes on board cars (which take up an area C of deck space each) and trucks (which take up an area T of deck space each). Each car pays $p for the crossing and each truck pays $q. Let x and y represent the number of cars and trucks on the deck of the ferry. (a) What are the variables and what are the parameters? Parameters are variables which are held constant during the modelling exercise. The variables vary during the modelling exercise. Parameters? A, C, T, p, q Variables? x, y

34. Modelling example 3 A ferry has a total deck space of area A. It takes on board cars (which take up an area C of deck space each) and trucks (which take up an area T of deck space each). Each car pays $p for the crossing and each truck pays $q. Let x and y represent the number of cars and trucks on the deck of the ferry. • Write down an expression for the revenue, R. • Write down a restriction on the values x and y due to thelimited deck space.

35. Modelling example 3 A ferry has a total deck space of area A. It takes on board cars (which take up an area C of deck space each) and trucks (which take up an area T of deck space each). Each car pays $p for the crossing and each truck pays $q. Let x and y represent the number of cars and trucks on the deck of the ferry. • List some assumptions made during the modelling process There are many possible assumptions, some such as “the earth will not be wiped out by an asteroid” while having some bearing on the problem will not affect the details of the mathematical working out. Your assumptions should be those that would affect the mathematical modelling of the problem if they were did not hold. They are often simplifications of reality Two possible assumptions are: • Unlimited cars & trucks are available • All of the available deck space is used

36. Modelling example 4 You are organising a meeting. The room costs $C to hire and you want to spend $F per person on food and drink. If n people attend(a) What is the total cost? (b) How much should each person be charged to break even? (c)If the price pre ticket is , how many people should attend to break even? Total revenue: Total cost: Break even: so

37. Summary Skills • Estimation • Construct equations • Know the structure of a report

38. Modelling example 3 A ferry has a total deck space of area A. It takes on board cars (which take up an area C of deck space each) and trucks (which take up an area T of deck space each). Each car pays $p for the crossing and each truck pays $q. Let x and y represent the number of cars and trucks on the deck of the ferry. • What are the variables and what are the parameters? • Write down an expression for revenue, R. • Write down a restriction on the numbers of cars and trucks. • List some assumptions.

39. Modelling example 4 You are organising a meeting. The room costs $C to hire and you want to spend $F per person on food and drink. If n people attend(a) What is the total cost? (b) How much should each person be charged to break even? (c)If the price pre ticket is , how many people should attend to break even?