1. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §1.4 Math Models

2. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §1.3 → Lines & LinearFunctions  Any QUESTIONS AboutHomeWork §1.3 → HW-03  h ≡ Si PreFix for 100X; e.g.: $100 = $h 100 Units = hU 1.3

3. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 3 Bruce Mayer, PE Chabot College Mathematics §1.4 Learning Goals  Study general modeling procedure  Explore a variety of applied models  Investigate market equilibrium and break-even analysis in economics

4. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 4 Bruce Mayer, PE Chabot College Mathematics Functional Math Modelling  Mathematical modeling is the process of translating statements in WORDS & DIAGRAMS into equivalent statements in mathematics. This Typically an ITERATIVE Process; the model is continuously adjusted to produce Real-World Results

5. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 5 Bruce Mayer, PE Chabot College Mathematics P1.4-10: Radium Decay Rate  A Sample of Radium (Ra) decays at a rate, R Ra, that is ProPortional to the amount of Radium, m Ra, Remaining  Express the Rate of Decay, R Ra, as a function of the ReMaining Amount, m Ra Ra Elemental Facts:

6. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 6 Bruce Mayer, PE Chabot College Mathematics

7. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 7 Bruce Mayer, PE Chabot College Mathematics Marketing of Products A & B  Profit Fcn given x% of Marketing Budget Spent on product A: a.Sketch Graph b.Find P(50) for 50-50 marketing expense c.Find P(y) where y is the % of Markeing Budget expended on Product B

8. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 8 Bruce Mayer, PE Chabot College Mathematics Marketing of Products A & B  Make T-Table to Sketch Graph  Note that only END POINTS of lines are needed to plot piece-wise Linear Function

9. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 9 Bruce Mayer, PE Chabot College Mathematics The Plot (By MATLAB)

10. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 10 Bruce Mayer, PE Chabot College Mathematics Profit for x = 50% 50 51

11. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 11 Bruce Mayer, PE Chabot College Mathematics

12. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 12 Bruce Mayer, PE Chabot College Mathematics

13. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 13 Bruce Mayer, PE Chabot College Mathematics Caveat Exemplars (Beware Models)  Q) From WHERE do these Math Models Come?  A) From PEOPLE; Including Me and YOU!  View Math Models with Considerable SKEPTISISM! Physical-Law Models are the Best Statistical Models (curve fits) are OK Human-Judgment Models are WORST

14. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 14 Bruce Mayer, PE Chabot College Mathematics Caveat Exemplars (Beware Models)  ALL Math Models MUST be verified against RealWorld RESULTS; e.g.: CFD (Physical) Models Checked by Wind Tunnel Testing at NASA-Ames Biology species-population models (curve- fits) tested against field observations Stock-Market Models are discarded often  LEAST Reliable models are those that depend on HUMAN BEHAVIOR (e.g. Econ Models) that can Change Rapidly

15. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 15 Bruce Mayer, PE Chabot College Mathematics P1.4-38 Greeting Card BreakEven  Make & Sell Greeting Cards Sell Price, S = $2.75/card Fixed Costs, C f = $12k Variable Costs, C v = $0.35/Card  Let x ≡ Number of Cards  Find Total Revenue, R(x) Total Cost, C(x) BreakEven Volume

16. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 16 Bruce Mayer, PE Chabot College Mathematics R & C Plot Break Even

17. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 17 Bruce Mayer, PE Chabot College Mathematics P & L Zones LOSS Zone Profit Zone

18. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 18 Bruce Mayer, PE Chabot College Mathematics MATLAB code % Bruce Mayer, PE % MTH-15 27Jun13 % M15_P14_38_Greeting_Card_Profit_Plot_1306.m % Ref: E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. % Herhold, G. C. Gregory, "An Engineer's Guide to MATLAB", ISBN % 978-0-13-199110-1, Pearson Higher Ed, 2011, pp294-295 % clc; clear % The Function xmin = 0; xmax = 8000; % in Cards ymin = 0; ymax = 22000 % in $; x = linspace(xmin,xmax,500); S = 2.75 % $k/card Cv = 0.35 % $/card Cf = 12000 % $ R = S*x; C = Cv*x + Cf; P = R - C; % % Use fzero to find Crossing Point Zfcn = @(u) S*u - (Cv*u + Cf) % Check Zereos by Plot y3 = Zfcn(x); plot(x, y3,[0,xmax], [0,0], 'LineWidth', 3),grid, title(['\fontsize{16}ZERO Plot',]) display('Showing fcn ZERO Plot; hit ANY KEY to Continue') pause % % Find Zeros xE = fzero(Zfcn,[4000 6000]) PE = S*xE - (Cv*xE + Cf) plot(x,R/1000, x,C/1000, 'k','LineWidth', 2), axis([0 xmax ymin ymax/1000]),... grid, xlabel('\fontsize{14}x (cards)'), ylabel('\fontsize{14}R&C ($k)'),... title(['\fontsize{16}MTH15 Bruce Mayer, PE P1.4-38',]),... annotation('textbox',[.41.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'M15P1438GreetingCardProfitPlot1306.m','FontSize',7) display('Showing 2Fcn Plot; hit ANY KEY to Continue') % "hold" = Retain current graph when adding new graphs hold on pause % xn = linspace(xmin, xmax, 100); fill([xn,fliplr(xn)],[S*xn/1000, fliplr(Cv*xn + Cf)/1000],'m')

19. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 19 Bruce Mayer, PE Chabot College Mathematics

20. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 20 Bruce Mayer, PE Chabot College Mathematics

21. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 21 Bruce Mayer, PE Chabot College Mathematics

22. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 22 Bruce Mayer, PE Chabot College Mathematics

23. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 23 Bruce Mayer, PE Chabot College Mathematics P1.4-60  Build a Box  Given 18” Square of CardBoard, then Construct Largest Volume Box 18” x x

24. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 24 Bruce Mayer, PE Chabot College Mathematics

25. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 25 Bruce Mayer, PE Chabot College Mathematics Largest Box 432

26. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 26 Bruce Mayer, PE Chabot College Mathematics MATLAB & MuPAD % Bruce Mayer, PE % MTH-15 23Jun13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % ref: % clear; clc % % The Limits xmin = 0; xmax = 9; ymin = 0; ymax = 450; % The FUNCTION x = linspace(xmin,xmax,500); y = x.*(18-2*x).^2; % % The ZERO Lines +> Do not need this time % * zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % FIND the Max Point Imax = find(y>=max(y)); Vmax = max(y), Xmax = x(Imax) % % the Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, Xmax,Vmax, 'p', 'LineWidth', 3),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}Box Height, x (inches)'), ylabel('\fontsize{14}Box Volume, V (inches^3)'),... title(['\fontsize{16}MTH15 Bruce Mayer, PE P1.4- 60',]),... annotation('textbox',[.51.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15P1460BoxConstructionVolume1306.m','FontSize',7) q := x*(18-x)^2 Simplify(q) expand(q)

27. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 27 Bruce Mayer, PE Chabot College Mathematics Surf Area Prob  Find the Surface Area for this Solid  Find By SUBTRACTION =  + NEW Exposed Surface

28. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 28 Bruce Mayer, PE Chabot College Mathematics Surf Area Prob cont.1  The Box Surf. Area  B = 4-Sides + [Top & Bot]  B = 4xh + 2x 2  The Cylinder Area  C = [Top & Bot] − Sides  C = 2πr 2 − π(2r)h

29. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 29 Bruce Mayer, PE Chabot College Mathematics Surf Area P cont.2  Then the NET Surface Area, S, by  S = B – C = [4xh + 2x 2 ] – [2πr 2 – π(2r)h] = 2x 2 – 2πr 2 + 2πrh + 4xh =+ S BC

30. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 30 Bruce Mayer, PE Chabot College Mathematics All Done for Today Fluid Mechanics Math Model

31. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 31 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

32. BMayer@ChabotCollege.edu MTH15_Lec-04_sec_1-4_Functional_Models_.pptx 32 Bruce Mayer, PE Chabot College Mathematics