1. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §3.5 Added Optimization

2. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §3.4 → Optimization & Elasticity  Any QUESTIONS About HomeWork §3.4 → HW-16 3.4

3. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 3 Bruce Mayer, PE Chabot College Mathematics §3.5 Learning Goals  List and explore guidelines for solving optimization problems  Model and analyze a variety of optimization problems  Examine inventory control

4. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 4 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics TransLate: Words → Math

5. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 5 Bruce Mayer, PE Chabot College Mathematics Applications Tips  The Most Important Part of Solving REAL WORLD (Applied Math) Problems  The Two Keys to the Translation LETUse the LET Statement to ASSIGN VARIABLES (Letters) to Unknown Quantities Analyze the RELATIONSHIP Among the Variables and Constraints (Constants)

6. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 6 Bruce Mayer, PE Chabot College Mathematics Basic Terminology  A LETTER that can be any one of various numbers is called a VARIABLE.  If a LETTER always represents a particular number that NEVER CHANGES, it is called a CONSTANT A & B are CONSTANTS

7. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 7 Bruce Mayer, PE Chabot College Mathematics Algebraic Expressions  An ALGEBRAIC EXPRESSION consists of variables, numbers, and operation signs. Some Examples  When an EQUAL SIGN is placed between two expressions, an equation is formed →

8. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 8 Bruce Mayer, PE Chabot College Mathematics Translate: English → Algebra  “Word Problems” must be stated in ALGEBRAIC form using Key Words per of less than more than ratio twicedecreased byincreased by quotient of times minus plus divided byproduct ofdifference of sum of divide multiply subtract add DivisionMultiplicationSubtractionAddition

9. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Translation  Translate this Expression: Eight more than twice the product of 5 and a number  SOLUTION LET n ≡ the UNknown Number Eight more than twice the product of 5 and a number.

10. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 10 Bruce Mayer, PE Chabot College Mathematics Mathematical Model  A mathematical model is an equation or inequality that describes a real situation.  Models for many applied (or “Word”) problems already exist and are called FORMULAS  A FORMULA is a mathematical equation in which variables are used to describe a relationship

11. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 11 Bruce Mayer, PE Chabot College Mathematics Formula Describes Relationship RelationshipMathematical Formula Perimeter of a triangle: a b c h Area of a triangle:

12. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Volume of Cone RelationshipMathematical Formulae h r Volume of a cone: Surface area of a cone:

13. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  °F ↔ °C RelationshipMathematical Formulae Celsius to Fahrenheit: Fahrenheit to Celsius: CelsiusFahrenheit

14. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Mixtures RelationshipMathematical Formula Percent Acid, P : Base Acid

15. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 15 Bruce Mayer, PE Chabot College Mathematics Solving Application Problems 1.Read the problem as many times as needed to understand it thoroughly. Pay close attention to the questions asked to help identify the quantity the variable(s) should represent. In other Words, FAMILIARIZE yourself with the intent of the problem Often times performing a GUESS & CHECK operation facilitates this Familiarization step

16. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 16 Bruce Mayer, PE Chabot College Mathematics Solving Application Problems 2.Assign a variable or variables to represent the quantity you are looking for, and, when necessary, express all other unknown quantities in terms of this variable. That is, Use at LET statement to clearly state the MEANING of all variables Frequently, it is helpful to draw a diagram to illustrate the problem or to set up a table to organize the information

17. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 17 Bruce Mayer, PE Chabot College Mathematics Solving Application Problems 3.Write an equation or equations that describe(s) the situation. That is, TRANSLATE the words into mathematical Equations 4.Solve the equation; i.e., CARRY OUT the mathematical operations to solve for the assigned Variables 5.CHECK the answer against the description of the original problem (not just the equation solved in step 4)

18. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 18 Bruce Mayer, PE Chabot College Mathematics Solving Application Problems 6.Answer the question asked in the problem. That is, make at STATEMENT in words that clearly addressed the original question posed in the problem description

19. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Mixture Problem  A coffee shop is considering a new mixture of coffee beans. It will be created with Italian Roast beans costing $9.95 per pound and the Venezuelan Blend beans costing $11.25 per pound. The types will be mixed to form a 60-lb batch that sells for $10.50 per pound.  How many pounds of each type of bean should go into the blend?

20. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans cont.2 1.Familiarize – This problem is similar to our previous examples. Instead of pizza stones we have coffee beans We have two different prices per pound. Instead of knowing the total amount paid, we know the weight and price per pound of the new blend being made.  LET  LET: i ≡ no. lbs of Italian roast and v ≡ no. lbs of Venezuelan blend

21. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans cont.3 2.Translate – Since a 60-lb batch is being made, we have i + v = 60. Present the information in a table. ItalianVenezuelanNew Blend Number of pounds iv60 Price per pound $9.95$11.25$10.50 Value of beans 9.95i11.25v630 i + v = 60 9.95i + 11.25v = 630

22. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans cont.4 2.Translate – We have translated to a system of equations 3.Carry Out – When equation (1) is solved for v, we have: v = 60  i. We then substitute for v in equation (2).

23. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans cont.5 3.Carry Out – Find v using v = 60  i. 4.Check – If 34.6 lb of Italian Roast and 25.4 lb of Venezuelan Blend are mixed, a 60-lb blend will result. The value of 34.6 lb of Italian beans is 34.6($9.95), or $344.27. The value of 25.4 lb of Venezuelan Blend is 25.4($11.25), or $285.75,

24. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Coffee Beans cont.6 4.Check – cont. so the value of the blend is [$344.27 + $285.75] = $630.02. A 60-lb blend priced at $10.50 a pound is also worth $630, so our answer checks 5.State – The blend should be made from 34.6 pounds of Italian Roast beans 25.4 pounds of Venezuelan Blend beans

25. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Max Enclosed Area  A rancher wants to build rectangular enclosures for her cows and horses. She divides the rectangular space in half vertically, using fencing to separate the groups of animals and surround the space.  If she has purchased 864 yards of fencing, what dimensions give the maximum area of the total space and what is the area of each enclosure?

26. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Max Enclosed Area  SOLUTION:  First Draw Diagram, Letting w ≡ Enclosure Width in yards l ≡ Enclosure Length in yards  Then the total Enclose Area for the large Rectangle

27. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  Max Enclosed Area  The fencing required for the enclosure is the perimeter of the rectangle plus the length of the vertical fencing between enclosures

28. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  Max Enclosed Area  Need to Maximize This Fcn:  However the fcn includes TWO UNknowns: length and width. Need to eliminate one variable (either one) in order to Product a function of one variable to maximize.  Use the equation for total fencing and isolate length l: Solving for l

29. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  Max Enclosed Area  Now we substitute the value for l into the area equation:  Maximize this function first by finding critical points by setting the first Derivative equal to Zero

30. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  Max Enclosed Area  Set dA/dw to zero, then solve  Since There is only one critical point, the Extrema at w = 144 is Absolute  Thus apply the second derivative test (ConCavity) to determine max or min

31. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example  Max Enclosed Area  Since d 2 A/dw 2 is ALWAYS Negative, then the A(w) curve is ConCave DOWN EveryWhere Thus a MAX exists at w = 144  Now find the length of the total space using our perimeter equation when solved for length

32. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example  Max Enclosed Area  Then The total space should be a 144yd by 216yd Rectangle. Each enclosure then is 144 yards wide and 216/2 - 108 yards long, and the area of each is 144yd·108yd = 15 552 sq-yd ↑ 144yd ↓ ← 216yd → ← 108yd → 15 552 yd 2

33. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  Find Minimum Cost  The daily production cost associated with a company’s principal product, the ChabotPad (or cPad), is inversely proportional to the length of time, in weeks, since the cPad’s release. Also, maintenance costs are linear and increasing.  At what time is total cost minimized? The answer may contain constants

34. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 34 Bruce Mayer, PE Chabot College Mathematics Example  Find Minimum Cost  SOLUTION:  Translate: at any given time, t,  Or  Now for C P → production cost associated with the cPad is inversely proportional to the length of time Formulaically TotalCost = ProductionCost + MaintenanceCost

35. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 35 Bruce Mayer, PE Chabot College Mathematics Example  Find Minimum Cost –t ≡ time in Weeks –K ≡ The Constant of ProPortionality in k$·weeks  Now for C M → maintenance costs are linear and increasing Translated to a Eqn –m ≡ Slope Constant (positive) in $k/week – b ≡ Intercept Constant (positive) in $k  Then, the Total Cost

36. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 36 Bruce Mayer, PE Chabot College Mathematics Example  Find Minimum Cost  find potential extrema by solving the derivative function set equal to zero:  Since t MUST be POSITIVE →

37. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 37 Bruce Mayer, PE Chabot College Mathematics Example  Find Minimum Cost  Now use the second derivative test for absolute extrema to verify that this value of t produces a positive ConCavity (UP) which confirm a minimum value for cost:  At the Zero Value

38. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 38 Bruce Mayer, PE Chabot College Mathematics Example  Find Minimum Cost  Since K and m are BOTH Positive then Is also Positive  The 2 nd Derivative Test Confirms that the function is ConCave UP at the zero point, which confirms the MINIMUM  The Min Cost:

39. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 39 Bruce Mayer, PE Chabot College Mathematics Example  Find Minimum Cost  STATE: for the cPad Minimum Total Cost will occur at this many weeks The Total Cost at this time in $k

40. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 40 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  Gonzalo walks west on a sidewalk along the edge of the grass in front of the Education complex of the University of Oregon.  The grassy area is 200 feet East-West and 300 feet North-South. Gonzalo strolls at 4 ft/sec on sidewalk 2 ft/sec on grass.  From the NE corner how long should he walk on the sidewalk before cutting diagonally across the grass to reach the SW corner of the field in the shortest time?

41. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 41 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  SOLUTION:  Need to TransLate Words to Math Relations  First DRAW DIAGRAM Letting: x ≡ The SideWalk Distance d ≡ The DiaGonal Grass Distance

42. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 42 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  The total distance traveled is x+d, and need to minimize the time spent traveling, so use the physical relationship [Distance] = [Speed]·[Time]  Solving the above “Rate” Eqn for Time:  So the time spent traveling on the SideWalk at 4 ft/s

43. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 43 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  Next, the time on Grass →  Writing in terms of x requires the use of the Pythagorean Theorem:  Then

44. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 44 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  And the Total Travel Time, t, is the SideWalk-Time Plus the Grass-Time  Now Set the 1 st Derivative to Zero to find t min

45. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 45 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  Continuing with the Reduction  OR

46. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 46 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  Using More Algebra

47. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 47 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  With Yet More Algebra  But the Diagram shows that x can NOT be more than 200ft, thus 26.79ft is the only relevant location of a critical point

48. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 48 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  Alternatives to check for max OR min: 2 nd Derivative Test –We could check using the second derivative test for absolute extrema to see if 26.79 corresponds to an absolute minimum, but that involves even more messy calculations beyond what we’ve already accomplished. Slope Value-Diagram and Direction- Diagram (Sign Charts) –Instead, check the critical point against the two endpoints on either side of x = 26.79; say x=0 & x=200

49. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 49 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  Find t-Value at x=0, x=26.79 and x=200

50. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 50 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  Find dt/dx Slope at x=0 and x=200

51. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 51 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time Value Summary Bar Chart  t is smallest at x = 26.79 Slope Summary Bar Chart  Slopes have different SIGNS on either side of x = 26.79

52. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 52 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 15Jul13 % % The Bar Values for x = [0 26.79 200] t = [180.3 179.9 200] % % the Bar Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green bar(t),axis([0 4 179 181]),... grid, xlabel('\fontsize{14}x = [0, 26.79, 200]'), ylabel('\fontsize{14}t(x) (sec)'),... title(['\fontsize{16}MTH15 t(x) Value-Chart',]),... annotation('textbox',[.15.8.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE 15Jul13 ','FontSize',7) % Bruce Mayer, PE % MTH-15 15Jul13 % % The Bar Values for x = [0 26.79 200] t = [-0.0273 0.25] % % the Bar Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green bar(t),axis([0 4 -.05.25]),... grid, xlabel('\fontsize{14}x = [0, 26.79, 200]'), ylabel('\fontsize{14}dt/dx (sec/ft)'),... title(['\fontsize{16}MTH15 dt/dx Slope-Chart',]),... annotation('textbox',[.15.8.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE 15Jul13 ','FontSize',7)

53. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 53 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  The T-Tables for the Value and Slope Diagrams  Both the Value & Slope Analyses confirm that x ≈ 26.79 is an absolute minimum  In other words, if Gonzalo walks on the sidewalk for about 26.79 feet and then walks directly to the southwest corner through the grass, he will spend the minimum time of about 179.90 seconds walking.

54. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 54 Bruce Mayer, PE Chabot College Mathematics Example  Minimize Travel Time  Plot by MuPAD

55. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 55 Bruce Mayer, PE Chabot College Mathematics MuPAD Code MuPAD Code Bruce Mayer, PE MTH15 15Jul13 MTH15_Minimize_Travel_Time_1307.mn t := x/4 + sqrt((200-x)^2+300^2)/2 t0 := subs(t, x = 0) float(t0) t200 := subs(t, x=200) dtdx := Simplify(diff(t,x)) u := solve(dtdx=0, x) float(u) dtdx0 := subs(dtdx, x = 0) float(dtdx0) tmin := subs(t, x = u) float(tmin) dtdx200 := subs(dtdx, x = 200) float(dtdx200) plot(t, x =0..200, GridVisible = TRUE, LineWidth = 0.04*unit::inch, plot::Scene2d::BackgroundColor = RGB::colorName([.8, 1, 1]), XAxisTitle = " x (ft) ", YAxisTitle = " t (s) " )

56. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 56 Bruce Mayer, PE Chabot College Mathematics MuPAD Code

57. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 57 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §3.5 P9 → GeoMetry + Calculus Special Prob → Enclosure Cost –Total Enclosed Area = 1600 ft 2 –Fence Costs in $/Lineal-Ft  Straight = 30  Curved = 40 See File → MTH15_Lec-17a_Fa13_sec_3- 5_Round_End_Fence_Enclosu re.pptx

58. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 58 Bruce Mayer, PE Chabot College Mathematics All Done for Today TravelTime or TimeTravel?

59. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 59 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

60. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 60 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart abc −−−−−−++++++−−−−−−++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

61. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 61 Bruce Mayer, PE Chabot College Mathematics

62. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 62 Bruce Mayer, PE Chabot College Mathematics

63. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 63 Bruce Mayer, PE Chabot College Mathematics

64. BMayer@ChabotCollege.edu MTH15_Lec-17_sec_3-5_Added_Optimization.pptx 64 Bruce Mayer, PE Chabot College Mathematics