Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

1. Chabot Mathematics §1.4 MathModels Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 1.3 Review § • Any QUESTIONS About • §1.3 → Lines & LinearFunctions • Any QUESTIONS AboutHomeWork • §1.3 → HW-03 • h ≡ Si PreFixfor 100X; e.g.: • $100 = $h • 100 Units = hU

3. §1.4 Learning Goals • Study general modeling procedure • Explore a variety of applied models • Investigate market equilibrium and break-even analysis in economics

4. 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. P1.4-10: Radium Decay Rate • A Sample of Radium (Ra) decays at a rate, RRa, that is ProPortional to the amount of Radium, mRa, Remaining • Express the Rate of Decay, RRa, as a function of the ReMaining Amount, mRa Ra Elemental Facts:

7. Marketing of Products A & B • Profit Fcn given x% of Marketing Budget Spent on product A: • Sketch Graph • Find P(50) for 50-50 marketing expense • Find P(y) where y is the % of Markeing Budget expended on Product B

8. 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. The Plot (By MATLAB)

10. Profit for x = 50% 51 50

13. 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. 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. P1.4-38 Greeting Card BreakEven • Make & Sell Greeting Cards • Sell Price, S = $2.75/card • Fixed Costs, Cf = $12k • Variable Costs, Cv = $0.35/Card • Let x ≡ Number of Cards • Find • Total Revenue, R(x) • Total Cost, C(x) • BreakEven Volume

16. R & C Plot Break Even

17. P & L Zones ProfitZone LOSSZone

18. % 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 xmaxyminymax/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') MATLAB code

23. P1.4-60  Build a Box • Given 18” Square of CardBoard, then Construct Largest Volume Box 18” x x

25. Largest Box 432

26. q := x*(18-x)^2 Simplify(q) expand(q) 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % 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([xminxmaxyminymax]),... 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)

27. Surf Area Prob • Find the SurfaceArea for this Solid • Find By SUBTRACTION  = + NEW Exposed Surface

28. The Box Surf. Area Surf Area Probcont.1 • The Cylinder Area • B = 4-Sides + [Top & Bot] • B = 4•xh + 2•x2 • C = [Top & Bot] − Sides • C = 2•πr2− π•(2r)•h

29. Surf Area P cont.2 • Then the NET Surface Area, S, by S B C = + • S = B – C = [4xh + 2x2] – [2•πr2–π•(2r)•h] = 2x2–2πr2 + 2πrh + 4xh

30. All Done for Today FluidMechanicsMath Model

31. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –