1. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §8.2 Quadratic Equation Apps

2. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §8.2 → Complete-the-Square  Any QUESTIONS About HomeWork §8.2 → HW-32 8.2 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 3 Bruce Mayer, PE Chabot College Mathematics §8.2 Quadratic Formula  The Quadratic Formula  Problem Solving with the Quadratic Formula

4. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 4 Bruce Mayer, PE Chabot College Mathematics The Quadratic Formula  The solutions of ax 2 + bx + c = 0 are given by This is one of the MOST FAMOUS Formulas in all of Mathematics

5. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Circular WalkWay  A Circular Pond 10 feet in diameter is to be surrounded by a “Paver” walkway that will be 2 feet wide. Find the AREA of the WalkWay  Familiarize: Recall the Formula for the area, A,of a Circle based on it’s radius, r  Also the diameter, d, is half of r. Thus A in terms of d

6. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Circular WalkWay  Familiarize: Make a DIAGRAM  Translate: Use Diagram of Subtractive Geometry = −

7. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Circular WalkWay  Translate: Diagram to Equation =−

8. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Circular WalkWay  CarryOUT: Solve Eqn for A walk  Using π ≈ 3.14 find

9. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Circular WalkWay  Check: Use A circle = πr 2  State: The Area of the Paver Walkway is about 75.4 ft 2 Note that UNITS must be included in the Answer Statement 

10. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Partition Bldg  A rectangular building whose depth (from the front of the building) is three times its frontage is divided into two parts by a partition that is 45 feet from, and parallel to, the front wall. If the rear portion of the building contains 2100 square feet, find the dimensions of the building.  Familiarize: REALLY needs a Diagram

11. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Partition Bldg  Familiarize: by Diagram  Now LET x ≡ frontage of building, in feet.  Translate: The other statements into Equations involving x

12. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Partition Bldg  The Bldg depth is three times its frontage, x → 3x = depth of building, in feet  The Bldg Depth is divided into two parts by a partition that is 45 feet from, and parallel to, the front wall → 3x – 45 = depth of rear portion, in ft

13. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Partition Bldg  Now use the 2100 ft 2 Area Constraint Area of rear = 2100 Area = x(3x−45), so

14. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Partition Bldg  So x is either 35ft or −20ft  But again Distances can NOT be negative  Thus x = 35 ft  Check: Use 2100 ft 2 Area 

15. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Partition Bldg  State:  The Bldg Frontage is 35ft  The Bldg Depth is 3(35ft) = 105ft 105’ 35’ 60’

16. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  Devon set out on a 16 mile Bike Ride. Unfortunately after 10 miles of Biking BOTH tires Blew Out. Devon Had to complete the trip on Foot.  Devon biked four miles per hour (4 mph) faster than she walked  The Entire journey took 2hrs and 40min Find Devon’s Walking Speed (or Rate)

17. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  Familiarize: Make diagram  LET w ≡ Devon’s Walking Speed  Recall the RATE Equation for Speed Distance = (Speed)·(Time)

18. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  Translate: The Biking Speed, b, is 4 mph faster than the Walking Speed →  From the Diagram note Distances by Rate Equation: Biking Distance = 10 miles = b·t bike Walking Distance = 6 miles = w·t walk

19. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  Translate: Now the Total Distance of 16mi is the sum of the Biking & Walking Distances →  From the Spd Eqn: Time = Dist/Spd

20. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  Translate: Thus by Speed Eqn:  Next, the sum of the Biking and Walking times is 2hrs & 40min = 2-2/3hr →

21. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  CarryOut: Clear Fractions in the last Eqn by multiplying by the LCD of 3·(w+4) ·w:

22. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  CarryOut: Divide the last Eqn by 8 to yield a Quadratic Equation  This Quadratic eqn does NOT factor so use Quadratic Formula with a = 1, b = −2, and c = −9

23. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  CarryOut: find w by Quadratic Formula  So Devon’s Walking Speed is

24. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  CarryOut: Since SPEED can NOT be Negative find:  Check: Test to see that the time adds up to 2.67 hrs 

25. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Bike Tire BlowOut  State: Devon’s Walking Speed was about 4.16 mph (quite a brisk pace)

26. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Golden Rectangle  Let’s Revisit the Derivation of the GOLDEN RATIO  A rectangle of length p and width q, with p > q, is called a golden rectangle if you can divide the rectangle into a square with side of length q and a smaller rectangle that is geometrically- similar to the original one.  The GOLDEN RATION then = p/q

27. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Golden Rectangle  Familiarize: Make a Diagram

28. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Golden Rectangle  Translate: Use Diagram

29. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Golden Rectangle  Carry Out: LET Φ ≡ Golden Ratio = p/q

30. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Golden Rectangle  Carry Out: Since both p & q are distances they are then both POSITIVE  Thus Φ = p/q must be POSITIVE  State: GOLDEN RATIO as defined by the Golden Rectangle

31. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Pythagorus  The hypotenuse of a right triangle is 52 yards long. One leg is 28 yards longer than the other. Find the lengths of the legs 1.Familarize. First make a drawing and label it. Let s = length, in yards, of one leg. Then s + 28 = the length, in yards, of the other leg. s + 28 s 52

32. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 32 Bruce Mayer, PE Chabot College Mathematics Pythagorean Triangle 2.Translate. We use the Pythagorean theorem: s 2 + (s + 28) 2 = 52 2 3.Carry Out. Identify the Quadratic Formula values a, b, & c s + 28 s 52 s 2 + (s + 28) 2 = 52 2 s 2 + s 2 + 56s + 784 = 2704 2s 2 + 56s − 1920 = 0 s 2 + 28s − 960 = 0

33. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 33 Bruce Mayer, PE Chabot College Mathematics Pythagorean Triangle 3.Carry Out: With s 2 + 28s − 960 = 0 Find: a = 1, b = 28, c = −960 Evaluate the Quadratic Formula

34. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 34 Bruce Mayer, PE Chabot College Mathematics Pythagorean Triangle 3.Carry Out: Continue Quadratic Eval s + 28 s 52

35. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 35 Bruce Mayer, PE Chabot College Mathematics Pythagorean Triangle 4.Check. Length cannot be negative, so −48 does not check. 5.State. One leg is 20 yards and the other leg is 48 yards. s + 28yd = 48 yds s = 20 yds 52 yds

36. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 36 Bruce Mayer, PE Chabot College Mathematics Vertical Ballistics  In Physics 4A, you will learn the general formula for describing the height of an object after it has been thrown upwards: Where –g ≡ the acceleration due to gravity  A CONSTANT = 32.2 ft/s 2 = 9.81 m/s 2 –t ≡ the time in flight, in s –v 0 ≡ the initial velocity in ft/s or m/s –h 0 ≡ the initial height in ft or m

37. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  X-Games Jump  In an extreme games competition, a motorcyclist jumps with an initial velocity of 80 feet per second from a ramp height of 25 feet, landing on a ramp with a height of 15 feet.  Find the time the motorcyclist is in the air.

38. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 38 Bruce Mayer, PE Chabot College Mathematics Example  X-Games Jump  Familiarize: We are given the initial velocity, initial height, and final height and we are to find the time the bike is in the air. Can use the Ballistics Formula  Translate: Use the formula with h = 15, v 0 = 80, and h 0 = 25.  Use h = 15, v 0 = 80, and h 0 = 25

39. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 39 Bruce Mayer, PE Chabot College Mathematics Example  X-Games Jump  Carry Out:  Use the Quadratic Formula to find t a = −16.1, b = 80 and c = 10

40. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 40 Bruce Mayer, PE Chabot College Mathematics Example  X-Games Jump  Carry Out:  Since Times can NOT be Negative t ≈ 5.09 seconds  Check: Check by substuting 5.09 for t in the ballistics Eqn. The Details are left for later  State: The MotorCycle Flight-Time is very nearly 5.09 seconds

41. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 41 Bruce Mayer, PE Chabot College Mathematics Example  Biking Speed  Kang Woo (KW to his friends) traveled by Bicycle 48 miles at a certain speed. If he had gone 4 mph faster, the trip would have taken 1 hr less. Find KW’s average speed

42. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 42 Bruce Mayer, PE Chabot College Mathematics Example  Biking Speed  Familiarize: In this can tabulating the information can help. Let r represent the rate, in miles per hour, and t the time, in hours for Kang Woo’s trip DistanceSpeedTime 48rt r + 4t – 1 Uses the Rate/Spd Eqn → Rate = Qty/Time

43. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 43 Bruce Mayer, PE Chabot College Mathematics Example  Biking Speed  Translate: From the Table Obtain two Equations in r & t and  Carry out: A system of equations has been formed. We substitute for r from the first equation into the second and solve the resulting equation

44. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 44 Bruce Mayer, PE Chabot College Mathematics Example  Driving Speed  Carry out: Next Clear Fractions Multiplying by the LCD

45. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 45 Bruce Mayer, PE Chabot College Mathematics Example  Driving Speed  Carry out: The Last Eqn is Quadratic in t:  Solve by Quadratic Forumula with: a = 1, b = −1, and c = −12

46. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 46 Bruce Mayer, PE Chabot College Mathematics Example  Driving Speed  Carry out: By Quadratic Formula  Since TIMES can NOT be NEGATIVE, then t = 4 hours  Return to one of the table Eqns to find r 12 mph.

47. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 47 Bruce Mayer, PE Chabot College Mathematics Example  Driving Speed  Check: To see if 12 mph checks, we increase the speed 4 mph to16 mph and see how long the trip would have taken at that speed:  The Answer checks a 3hrs is indeed 1hr less than 4 hrs that the trip actually took  State: KW rode his bike at an average speed of 12 mph

48. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 48 Bruce Mayer, PE Chabot College Mathematics ReCall The WORK Principle  Suppose that A requires a units of time to complete a task and B requires b units of time to complete the same task.  Then A works at a rate of 1/a tasks per unit of time.  B works at a rate of 1/b tasks per unit of time,  Then A and B together work at a rate of [1/a + 1/b] per unit of time.

49. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 49 Bruce Mayer, PE Chabot College Mathematics The WORK Principle  If A and B, working together, require t units of time to complete a task, then their rate is 1/t and the following equations hold:

50. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 50 Bruce Mayer, PE Chabot College Mathematics Example  Empty Tower Tank  A water tower has two drainpipes attached to it. Working alone, the smaller pipe would take 20 minutes longer than the larger pipe to empty the tower. If both drainpipes work together, the tower can be drained in 40 minutes.  How long would it take the small pipe, working alone, to drain the tower?

51. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 51 Bruce Mayer, PE Chabot College Mathematics Example  Empty Tower Tank  Familiarize: Creating a Table helps to clarify the given data PipeTime to Complete the Job Alone Work Rate Combined Working Time Portion of Job Completed Smallert + 20 min.40 Largert min.40  And the Job-Portions must add up to ONE complete Job:

52. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 52 Bruce Mayer, PE Chabot College Mathematics Example  Empty Tower Tank  Carry Out:  The Last Eqn is Quadratic

53. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 53 Bruce Mayer, PE Chabot College Mathematics Example  Empty Tower Tank  Use Quadratic Formula:

54. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 54 Bruce Mayer, PE Chabot College Mathematics Example  Empty Tower Tank  Omit the negative solution as times cannot be negative  The amount of time required by the small pipe is represented by t + 20, it would take approximately 20 + 71.2 or 91.2 minutes  Check: Use the Work Eqn from Table 

55. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 55 Bruce Mayer, PE Chabot College Mathematics Example  Empty Tower Tank  State: Working alone the SMALL pipe would empty the Water Tower in about 91.2 minutes

56. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 56 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §8.2 Exercise Set 74  The Arrhenius Rate Equation

57. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 57 Bruce Mayer, PE Chabot College Mathematics All Done for Today MotorCyle Fatality Statistics

58. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 58 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

59. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 59 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

60. BMayer@ChabotCollege.edu MTH55_Lec-50_sec_8-2b_Quadratic_Eqn_Apps.ppt 60 Bruce Mayer, PE Chabot College Mathematics