1. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §6.7 Rational Eqn Apps

2. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §6.6 → Rational Equations  Any QUESTIONS About HomeWork §6.6 → HW-22 6.6 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 3 Bruce Mayer, PE Chabot College Mathematics §6.7 Rational Equation Applications  Problems Involving Work  Problems Involving Motion  Problems Involving Proportions  Problems involving Average Cost

4. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 4 Bruce Mayer, PE Chabot College Mathematics Solve a Formula for a Variable  Formulas occur frequently as mathematical models. Many formulas contain rational expressions, and to solve such formulas for a specified letter, we proceed as when solving rational equations.

5. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 5 Bruce Mayer, PE Chabot College Mathematics Solve Rational Eqn for a Variable 1.Determine the DESIRED letter (many times formulas contain multiple variables) 2.Multiply on both sides to clear fractions or decimals, if that is needed. 3.Multiply if necessary to remove parentheses. 4.Get all terms with the letter to be solved for on one side of the equation and all other terms on the other side, using the addition principle. 5.Factor out the unknown. 6.Solve for the letter in question, using the multiplication principle.

6. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Solve for Letter  Solve this formula for y:  SOLN: Multiplying both sides by the LCD Simplifying Dividing both sides by Ra Multiplying Subtracting RT

7. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Mechanics  In a hydraulic system, a fluid is confined to two connecting chambers. The pressure in each chamber is the same and is given by finding the force exerted (F) divided by the surface area (A). Therefore, we know  Solve this Eqn for A 2

8. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Mechanics  SOLUTION:  This formula can be used to calculate A 2 whenever A 1, F 2, and F 1 are known Multiplying both sides by the LCD Dividing both sides by F 1

9. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 9 Bruce Mayer, PE Chabot College Mathematics Problems Involving Work  Rondae and Marrisa work during the summer painting houses. Rondae can paint an average size house in 12 days Marrisa requires 8 days to do the same painting job.  How long would it take them, working together, to paint an average size house?

10. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 10 Bruce Mayer, PE Chabot College Mathematics House Painting cont. 1.Familiarize. We familiarize ourselves with the problem by exploring two common, but incorrect, approaches. a)One common, incorrect, approach is to add the two times. → 12 + 8 = 20 b)Another incorrect approach is to assume that Rondae and Marrisa each do half the painting. –Rondae does ½ in 12 days = 6 days –Marrisa does ½ in 8 days = 4 days –6 days + 4 days = 10 days.  

11. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 11 Bruce Mayer, PE Chabot College Mathematics House Painting cont.  A correct approach is to consider how much of the painting job is finished in ONE day; i.e., consider the work RATE  It takes Rondae 12 days to finish painting a house, so his rate is 1/12 of the job per day.  It takes Marrisa 8 days to do the painting alone, so her rate is 1/8 of the job per day.  Working together, they can complete 1/8 + 1/12, or 5/24 of the job in one day.

12. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 12 Bruce Mayer, PE Chabot College Mathematics House Painting cont.  Note That given a TIME-Rate [ Amount ] = [ Rate ][ TimeQuantity ] t/8t1/8 Marrisa t/12t1/12 Rondae Amount Completed TimeRate of Work Painter  Form a table to help organize the info:

13. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 13 Bruce Mayer, PE Chabot College Mathematics House Painting cont. 2.Translate. The time that we want is some number t for which Portion of work done by Marrisa in t days Portion of work done by Rondae in t days Or Portion of work done together in t days

14. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 14 Bruce Mayer, PE Chabot College Mathematics House Painting cont. 3.Carry Out. We can choose any one of the above equations to solve:

15. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 15 Bruce Mayer, PE Chabot College Mathematics House Painting cont. 4.Check. Test t = 24/5 days  5.State. Together, it will take Rondae & Marrisa 4 & 4/5 days to complete painting a house.

16. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 16 Bruce Mayer, PE Chabot College Mathematics 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 total rate of [1/a + 1/b] per unit of time.

17. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 17 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 combined rate is 1/t and the following equations hold:

18. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 18 Bruce Mayer, PE Chabot College Mathematics Problems Involving Motion  Because of a tail wind, a jet is able to fly 20 mph faster than another jet that is flying into the wind. In the same time that it takes the first jet to travel 90 miles the second jet travels 80 miles. How fast is each jet traveling? r r+20  HEAD Wind  TAIL Wind

19. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 19 Bruce Mayer, PE Chabot College Mathematics HEADwind vs. TAILwind 1.Familiarize. We try a guess. If the fast jet is traveling 300 mph because of a tail wind the slow jet plane would be traveling 300−20 or 280 mph. At 300 mph the fast jet would have a 90 mile travel-time of 90/300, or 3/10 hr. At 280 mph, the other jet would have a travel-time of 80/280 = 2/7 hr.  Now both planes spend the same amount of time traveling, So the guess is INcorrect.

20. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 20 Bruce Mayer, PE Chabot College Mathematics HEADwind vs. TAILwind 2.Translate. Fill in the blanks using [ TimeQuantity ]=[ Distance ]/[ Rate ] Air Craft Distance (miles) Speed or Rate (miles per hour) Time (hours) Jet 180r Jet 290r + 20 r

21. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 21 Bruce Mayer, PE Chabot College Mathematics HEADwind vs. TAILwind  Set up a RATE Table [ Distance ]/[ Rate ] = [ TimeQuantity ] Air Craft Distance (miles) Speed (miles per hour) Time (hours) Jet 180r80/r Jet 290r + 2090/(r + 20) The Times MUST be the SAME

22. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 22 Bruce Mayer, PE Chabot College Mathematics HEADwind vs. TAILwind  Since the times must be the same for both planes, we have the equation 3.Carry Out. To solve the equation, we first Clear-Fractions multiplying both sides by the LCD of r(r+20)

23. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 23 Bruce Mayer, PE Chabot College Mathematics HEADwind vs. TAILwind  Complete the “Carry Out” Simplified by Clearing Fractions Using the distributive law Subtracting 80r from both sides Dividing both sides by 10  Now we have a possible solution. The speed of the slow jet is 160 mph and the speed of the fast jet is 180 mph

24. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 24 Bruce Mayer, PE Chabot College Mathematics HEADwind vs. TAILwind 4.Check. ReRead the problem to confirm that we were able to find the speeds. At 160 mph the jet would cover 80 miles in ½ hour and at 180 mph the other jet would cover 90 miles in ½ hour. Since the times are the same, the speeds Chk 5.State. One jet is traveling at 160 mph and the second jet is traveling at 180 mph

25. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 25 Bruce Mayer, PE Chabot College Mathematics Formulas in Economics  Linear Production Cost Function Where –b is the fixed cost in $ –a is the variable cost of producing each unit in $/unit (also called the marginal cost)  Average Cost ($/unit)

26. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 26 Bruce Mayer, PE Chabot College Mathematics Formulas in Economics  Price-Demand Function: Suppose x units can be sold (demanded) at a price of p dollars per units. Where –m & n are SLOPE Constants in $/unit & unit/$ –d & k are INTERCEPT Constants in $ & units

27. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 27 Bruce Mayer, PE Chabot College Mathematics Formulas in Economics  Revenue Function Revenue = (Price per unit)·(No. units sold)  Profit Function Profit = (Total Revenue) – (Total Cost)

28. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Average Cost  Metro Entertainment Co. spent $100,000 in production costs for its off-Broadway play Pride & Prejudice. Once running, each performance costs $1000 a)Write the Cost Function for conducting z performances b)Write the Average Cost Function for the z performances c)How many performances, n, result in an average cost of $1400 per show

29. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Average Cost  SOLUTION a) Total Cost is the sum of the Fixed Cost and the Variable Cost  SOLUTION b) The Average Cost Fcn

30. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Average Cost  SOLUTION c) In this case for “n” Shows  Thus 250 shows are needed to realize a per-show cost of $1400

31. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 31 Bruce Mayer, PE Chabot College Mathematics Problems Involving Proportions  Recall that a RATIO of two quantities is their QUOTIENT. For example, 45% is the ratio of 45 to 100, or 45/100.  A proportion is an equation stating that two ratios are EQUAL: An equality of ratios, A/B = C/D, is called a PROPORTION. The numbers within a proportion are said to be proportionAL to each other

32. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 32 Bruce Mayer, PE Chabot College Mathematics Example  Triangle Proportions  Triangles ABC and XYZ are “similar” A B C X Y Z a = 7 b x = 8 y = 12  Now Solve for b if x = 8, y = 12 and a = 7 Note that “Similar” Triangles are “In Proportion” to Each other

33. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 33 Bruce Mayer, PE Chabot College Mathematics Example  Similar Triangles  Set Up The Proportions A B C X Y Z a = 7 b x = 8 y = 12 [b is to 12] as [7 is to 8]

34. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 34 Bruce Mayer, PE Chabot College Mathematics Example  Similar Triangles  Alternative Proportions A B C X Y Z a = 7 b x = 8 y = 12 [b is to 7] as [12 is to 8]

35. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  Quantity Proportions  A sample of 186 hard drives contained 4 defective drives. How many defective drives would be expected in a group of 1302 HDDs?  Form a proportion in which the ratio of defective hard drives is expressed in 2 ways. defective drives total drives defective drives total drives  Expect to find 28 defective HDDs

36. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 36 Bruce Mayer, PE Chabot College Mathematics Whale Proportionality  To determine the number of humpback whales in a pod, a marine biologist, using tail markings, identifies 35 members of the pod.  Several weeks later, 50 whales from the SAME pod are randomly sighted. Of the 50 sighted, 18 are from the 35 originally identified. Estimate the number of whales in the pod.

37. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 37 Bruce Mayer, PE Chabot College Mathematics Tagged Whale Proportions 1.Familarize. We need to reread the problem to look for numbers that could be used to approximate a percentage of the of the pod sighted.  Since 18 of the 35 whales that were later sighted were among those originally identified, the ratio 18/50 estimates the percentage of the pod originally identified.

38. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 38 Bruce Mayer, PE Chabot College Mathematics HumpBack Whales 2.Translate: Stating the Proportion Marked whales sighted later Total Whales sighted later Whales originally Marked Total Whales in pod 3.Carry Out

39. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 39 Bruce Mayer, PE Chabot College Mathematics More On Whales 4.Check. The check is left to the student. 5.State. There are about 97 whales in the Pod

40. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 40 Bruce Mayer, PE Chabot College Mathematics One More Whale  Another way to summarize the RANDOM-Tagging and RANDOM-Sighting Relation: [35 is to w] as [18 is to 50]  Thus the Proportionality:  Solve for w

41. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 41 Bruce Mayer, PE Chabot College Mathematics Example  Vespa Scooters  Juan’s new scooter goes 4 mph faster than Josh does on his scooter. In the same time that it takes Juan to travel 54 miles, Josh travels 48 miles.  Find the speed of each scooter.

42. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 42 Bruce Mayer, PE Chabot College Mathematics Example  Vespa Scooters  Familiarize. Let’s guess that Juan is going 20 mph. Josh would then be traveling 20 – 4, or 16 mph.  At 16 mph, he would travel 48 miles in 3 hr. Going 20 mph, Juan would cover 54 mi in 54/20 = 2.7 hr. Since 3  2.7, our guess was wrong, but we can see that if r = the rate, in miles per hour, of Juan’s scooter, then the rate of Josh’s scooter = r – 4.

43. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 43 Bruce Mayer, PE Chabot College Mathematics Example  Vespa Scooters  LET: r ≡ Speed of Juan’s Scooter t ≡ The Travel Time for Both Scooters  Tabulate the data for clarity DistanceSpeedTime Juan’s Scooter Josh’s Scooter

44. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 44 Bruce Mayer, PE Chabot College Mathematics Example  Vespa Scooters  Translate. By looking at how we checked our guess, we see that in the Time column of the table, the t’s can be replaced, using the formula Time = Distance/Speed DistanceSpeedTime Juan’s Scooter Josh’s Scooter

45. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 45 Bruce Mayer, PE Chabot College Mathematics Example  Vespa Scooters  Since the Times are the SAME, then equate the two Time entries in the table as:  Carry Out

46. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 46 Bruce Mayer, PE Chabot College Mathematics Example  Vespa Scooters  Check: If our answer checks, Juan’s scooter is going 36 mph and Josh’s scooter is going 36 − 4 = 32 mph. Traveling 54 miles at 36 mph, Juan is riding for 54/36 or 1.5 hours. Traveling 48 miles at 32 mph, Josh is riding for 48/32 or 1.5 hours. The answer checks since the two times are the same.  State: Juan’s speed is 36 mph, and Josh’s speed is 32 mph

47. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 47 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §6.7 Exercise Set 16 (ppt), 34, 44  Mass Flow Rate for a Diverging Nozzle

48. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 48 Bruce Mayer, PE Chabot College Mathematics P6.7-16  Given Avg Cost Function Graph:  Find Production Quatity for Avg Cost of $425/Chair  SOLUTION: Cast Right & Down 20k  ANS → 20k Chairs/mon

49. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 49 Bruce Mayer, PE Chabot College Mathematics All Done for Today Human Proportions: HeadLength BaseLine

50. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 50 Bruce Mayer, PE Chabot College Mathematics Example  Similar Triangles  SOLUTION  Examine the drawing, write a proportion, and then solve. A B C X Y Z a = 7 b x = 8 y = 12  Note that side a is always opposite angle A, side x is always opposite angle X, and so on.

51. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 51 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

52. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 52 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

53. BMayer@ChabotCollege.edu MTH55_Lec-35_sec_6-7_Rational_Equation_Applications.ppt 53 Bruce Mayer, PE Chabot College Mathematics