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

Chabot Mathematics. §5.6 Int Apps Life&Soc Sci. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Final Exam Ref Document. Students may HAND WRITE at 3x5 CARD as open Reference for the Final Exam Final Exam Tu /17Dec13 at 6:30pm in Rm1613. 5.5. Review §.

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

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  1. Chabot Mathematics §5.6 Int AppsLife&SocSci Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. Final Exam Ref Document • Students may HAND WRITE at 3x5 CARD as open Reference for the Final Exam • Final Exam Tu/17Dec13at 6:30pmin Rm1613

  3. 5.5 Review § • Any QUESTIONS About • §5.5 → Biz & Econ Integral Apps • Any QUESTIONS About HomeWork • §5.5 → HW-26

  4. §5.6 Learning Goals • Examine survival and renewal functions • Use definite integration to compute • population-totals from Population Density • explore the flow of blood through an artery • Derive an integration formula for the volume of a solid of revolution, and use it to estimate the size of a tumor

  5. Survival & Renewal • Consider a Population of Spotted Yellow Squirrels (SYS) confined to a game preserve • The SYS are carefully counted every 5 years by Dept of Fish & Game Biologists. The Last “census” ended today with a total, P0, of 7500 SYS • The Biologists need a method of Estimating the change in Population before the next Census

  6. Survival & Renewal • After Researching the SYS the biologists have found • That the SYS have a LifeSpan (maximum age, Amax) of about 2200 days (≈ 6 yrs) • The SYS have a “Survival” function: • Where • t ≡ time in days • τ ≡ The “Time Constant” in % of MaxAge • In this case the time constant is 21.715% of MaxAge

  7. Survival and Renewal • The Survival fcn for the SYS

  8. % Bruce Mayer, PE % MTH-15 • 23Jun13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % clear; clc; clf; % clf clears figure window % % The Limits xmin = 0; xmax = 100; ymin = 0; ymax = 100; % in PerCent of MaxAge % The FUNCTION tau = 21.715 % in PerCent of MaxAge x = linspace(xmin,xmax,500); y = 100*exp(-x/tau); % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, 'LineWidth', 4),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14} Age, A (% of Max)'), ylabel('\fontsize{14}Survival %, S(A)'),... title(['\fontsize{16}MTH15 • Spotted Yellow Squirel',]),... annotation('textbox',[.21 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 27Jul113','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:10:xmax]); set(gca,'YTick',[ymin:10:ymax]) MATLAB Code

  9. Survival & Renewal • After Researching the SYS the biologists find That the SYS have a roughly constant Birth, or Replacement, Rate: • It is now 2 yrs after the Last census, so the “Term” of Projection, T, is 730 days • The Biologist can now develop a model for the Population, P(T) at T = 730 days

  10. Survival & Renewal • P(T) model Development • Multiply starting population by S(730) to determine how many of the original 7500 are alive 2 years later: • To N0S add the births over some short time period, ∆t, that survive until the end of the term • For example, it is much more likely that a SYS born on day 701 will survive as compared with a SYS born on day 49

  11. Survival and Renewal • P(T) model Development • In this case it is convenient to take∆t = 1 day. • Thus the number of SYS born on, say, day 440 that make it to day 730 must survive a total of 730−440 = 290 days; a math expression: % of those born on day 440 that survive to day 730 No. added on day 440 thatSurvive to 730 No. Born on day 440 = Rate·∆time

  12. Survival and Renewal • P(T) model Development • The No. Added by births in variable form • Then the Total SYS 2 years later • Recall that

  13. Survival & Renewal • P(T) model Development • Using • Rewrite P(T) as • Recognize that sum is in the Riemann form; Thus as ∆t→0, the Sum→Integral

  14. Survival & Renewal • P(T) model Development • So the final Math model if S(t) and R(t) are known: • Now can calc the SYS Population 2 years (730 d) after the last SYS-Count • Note: Times expressed as the % of Term-Time of 730days

  15. Survival & Renewal • Running the Numbers on MuPAD find • P(730) = 9483Spotted YellowSquirells • 5 Years Laterexpect aPopulation of about 10,032SYS MTH15_Spotted_Yellow_Squirel_S-n-R_1307.mn

  16. Example  RainBow Trout S&R • About 48% of rainbow trout stocked as fingerlings in the Clinch River die each year, so that the fraction surviving out to t years is e−0.65t • The stocking rate of new fish is about 50,000 per year. • If there are initially 63,000 fingerlings, how many are projected to remain after five years?

  17. Example  RainBow Trout S&R • SOLUTION: • This is a survival and renewal application, with • Then the number of fish present (in k-fish) after five years is given by

  18. Example  RainBow Trout S&R • Thus: • Now Engage the substitution • Find

  19. Example  RainBow Trout S&R • Running the Numbers • There are a projected 76,383 rainbow trout fingerlings in the river after 5 years

  20. Incremental area, dA = [Length]·[Width]→ Population Density • Calculate Population density (people divided by area) usingConcentric ring Integration People Living in the Ring = [Pop-Density at Location r]·[Area]

  21. Incremental area, dA = [Length]·[Width]→ Population Density • Then Add-up, or Integrate, all the dp’s to obtain the total no. of people living in the area 2πR People Living in the Ring = [Pop-Density at Location r]·[Area

  22. Population Density • Or • Or in condensed terms

  23. Example  Urban Population • A town’s population is centralized and drops off dramatically toward the outskirts of the city. • Census results suggest a model for the population density in k-People/mi2 • How many people are between two and three miles from the center of the city?

  24. Example  Urban Population • SOLUTION: • The population in the 2-3 mile ring: • Use Substitution

  25. Example  Urban Population • Making theSubstitutions • STATE: The population between 2&3 miles away from the center is approximately 13,065 People

  26. Volumes of Revolution • Rotate y = f(x) about x-axis form solid • At position xj the height of the disk r = y = f(xj ) • The Area of the disk at xj is the area of a circle, A = πr2 = π[f(xj )]2

  27. Volumes of Revolution • Rotate y = f(x) about x-axis form solid • Then the increment volume dV is the [DiskArea]·[DiskWidth] =dV = {π[f(xj )] 2} ·{dx} • Adding up all the Incremental Disk Volumes

  28. Example  Solid of Revolution • Find the volume of the solid created by rotating about the x-axis over x = [0,1] the Graph of

  29. Example  Solid of Revolution • The solid after Rotation

  30. Example  Solid of Revolution • SOLUTION: • Using the volume Formula: • The volume of the solid is approximately 0.239 cubic units

  31. WhiteBoard Work • Problems From §5.6 • P24 → Arterial Blood Flow • P36 → LifeExpectancy • P40 → HumanRespiration

  32. MATLAB Code % Bruce Mayer, PE % MTH-15 • 23Jun13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % clear; clc; clf; % clf clears figure window % % The Limits xmin = 0; xmax = 2.25; ymin = 0; ymax = 5; % in PerCent of MaxAge % The FUNCTION tau = 21.715 % in PerCent of MaxAge x = linspace(xmin,xmax,500); y = x.*(-1.2*x.^2 +5.72); Xint = roots([-1.2 0 5.72]) % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green area(x,y, 'LineWidth', 4, 'FaceColor',[0.6 0.8 1]),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14} Time, t (sec)'), ylabel('\fontsize{14} Respiration Flow, R (liter/sec)'),... title(['\fontsize{16}MTH15 • Human Respiration',]),... annotation('textbox',[.21 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 27Jul113','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:.25:xmax]); set(gca,'YTick',[ymin:1:ymax]) set(gca,'Layer','top')

  33. Xint = 2.1833

  34. All Done for Today A Rotated“Logistic”Curve

  35. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

  36. Bruce Mayer, PE MTH15 • 27Jul13 P5.6-36 L := 41.6*(1+1.07*t)^0.13 Lavg := (1/60)*int(L, t=10..70) T80 := subs(L, t = 80) T := 73.4916 Le := (1/T)*int(L, t=0..T) plot(L, t =0..100, GridVisible = TRUE,LineWidth = 0.04*unit::inch) y = Life Expectancy, L * t = Current Age From MATLAB >> Tz = @(T) 41.6*(1+1.07*T).^0.13 - T Tz = @(T)41.6*(1+1.07*T).^0.13-T >> LL = fzero(Tz, 50) LL = 73.4916

  37. Max @ (1.26, 4.807) Vtot = Area-Under-Curve Xint = 2.1833

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