1. MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Sections 7.6 – Applications and Variations  Motion (rate – time – distance)  Shared Work  Variation (direct, inverse, & joint) Copyright © 2010 by Ron Wallace, all rights reserved.

2. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water?

3. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt

4. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream

5. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6

6. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4

7. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t

8. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t Downstream

9. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t Downstream 12

10. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t Downstream 12r + 4

11. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t Downstream 12r + 4t

12. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t = 6/(r – 4) Downstream 12r + 4t = 12/(r + 4)

13. Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Time Up = Time Down

14. Motion In general … begin by filling in the table... Use a formula to eliminate a variable. Set equal expressions equal to each other. Solve & Check Motion drt Motion 1 ??? Motion 2 ???

15. Motion – Example A doctor drove 200 miles to attend a national convention. Because of poor weather, her average speed on the return trip was 10 mph less than her average speed going to the conventions. If the return trip took 1 hour longer, how fast did she drive in each direction? Motion drt Going there Coming home

16. Motion – Example A doctor drove 200 miles to attend a national convention. Because of poor weather, her average speed on the return trip was 10 mph less than her average speed going to the conventions. If the return trip took 1 hour longer, how fast did she drive in each direction? Motion drt Going there 200rt 1 = 200/r Coming home 200r – 10t 2 = 200/(r – 10)

17. Shared Work Example … Tom & Sue work for the city parks department where they mow the lawn in the city park. Tom, working by himself, can mow the lawn in 5 hours. Sue, working by herself, can mow the lawn in 4 hours. How long will it take to mow the lawn if they work together? Estimates?  9 hours?  4.5 hours?  2.25 hours?  Other guesses?

18. Tom (working alone) takes 5 Hours

19. Hour #1

20. Tom (working alone) takes 5 Hours Hour #1 Hour #2

21. Tom (working alone) takes 5 Hours Hour #1 Hour #2 Hour #3

22. Tom (working alone) takes 5 Hours Hour #1 Hour #2 Hour #3 Hour #4

23. Tom (working alone) takes 5 Hours Hour #1 Hour #2 Hour #3 Hour #4 Hour #5

24. Sue (working alone) takes 4 Hours

25. Hour #1

26. Sue (working alone) takes 4 Hours Hour #1 Hour #2

27. Sue (working alone) takes 4 Hours Hour #1 Hour #2 Hour #3

28. Sue (working alone) takes 4 Hours Hour #1 Hour #2 Hour #3 Hour #4

29. Working Together Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour.

30. Working Together Hour #1 Hour #1 Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour.

31. Working Together Hour #1 Hour #1 Hour #2 Hour #2 Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour.

32. Working Together Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour. Hour #1 Hour #1 Hour #2 Hour #2 Hour #3 ?

33. Working Together Let x = # hours to complete the job together. Portion of work completed by Tom Portion of work completed by Sue Adding these gives … Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour.

34. Shared Work – Summary

35. Proportions – Equality of Ratios Ratio:  A quotient of related quantities. Proportion:  Two equivalent rations that relate the same quantities.  One of the quantities will be unknown. Does it matter which quantity is on top?

36. Proportions – Equality of Ratios Example … An automobile gets 23 miles per gallon of gas (mpg). How much gas does it take to travel 200 miles? The Ratio …

37. Proportions – Equality of Ratios Example … An automobile gets 23 miles per gallon of gas (mpg). How much gas does it take to travel 200 miles? The Ratio …

38. Two Common Proportions Similar Triangles ◦ Corresponding Angles are Equal ◦ Ratios of Corresponding Sides are Equal Scale Drawings ◦ Maps ◦ Blueprints

39. Variation Direct Variation  Two quantities whose ratio is a constant. Inverse Variation  Two quantities whose product is a constant. Others – Combinations of the Above ◦ e.g. Joint Variation  Three quantities where the ratio of one of the quantities to the product of the other two quantities is a constant.

40. Direct Variation Two quantities whose ratio is a constant. “ y varies directly as x ”  aka: “ y is [directly] proportional to x ” k is called the “constant of proportionality” In an application, data is given to determine k and then values of x are used to determine values of y.

41. Direct Variation Example... The weight hanging from a spring is directly proportional to the distance the spring is stretched (Hooke’s Law). If a 6 pound weight stretches a particular spring 5 inches, and a fish hanging from the same spring stretches the spring 9 inches, how much does the fish weigh?

42. Inverse Variation Two quantities whose product is a constant. “ y varies inversely as x ”  aka: “ y is inversely proportional to x ” k is called the “constant of proportionality” In an application, data is given to determine k and then values of x are used to determine values of y.

43. Inverse Variation Example... The time it takes a to get sunburned varies inversely with the UV rating on that day? If a UV rating of 4 causes person with fair skin to burn in 20 minutes, how long will it take for them to burn on a day with a UV rating of 7? http://www.revolutionhealth.com/articles/uv-index/stu3205