Unit 3: Rational Expressions

Transcription

1 Unit : Rational Epressions Common Denominators Directions: For each of the following, practice finding the LCM necessary for creating a common denominator (Hint: make sure to factor). 1) ; 0. 14; 1 10 y ) 5; ; 1 5) 5 14 ;

2 Adding and Subtracting Rational Epressions Directions: Simplify each of the following rational epressions using the rules for adding and subtracting. 1) y y 4 10 ) y y ) 9 6 b b b 4) 10 5 a a a a a a 5) )

3 Adding and Subtracting Rational Epressions Homework: Eamples: Simplify and state the values for that result in the epression being undefined y ( y 6) 4 ( 7 8) ( y 6) ( y 8y 1) 1( 64)

4 4 Multiplying and Dividing Eamples: Simplify Multiply and Divide Homework: Simplify w z z z w z y y s t sr r s t y b y ab z w q p z w pq y y y y y y 9 1 a a a a a

5 5

6 Mied practice simplifying rational epressions

7

8

9 To Simplify Comple Fractions: Eamples: Simplify ( ) y y 9

10 Simplfying Comple Fractions Homework: 10

11 Simplifying Comple Fractions Day () 11

12

13 ( ) 1

14 Direct, Joint and Inverse Variation Direct Variation: Eamples: If y varies directly as and y = 1 when = -, find y when = 16. The number of centimeters y in a linear measurement varies directly with the number of inches in the measurement. Pablo s height is 15.4 centimeters or 60 inches. What is Maria s height in centimeters if she is 64 inches tall? Joint Variation: Eample: Suppose y varies jointly as and z. Find y when = 8 & z =, if y = 16 when z = & = 5. The cost c of materials for a deck varies jointly with the width w and the length l. If c = $ when w = 1 and l = 16, find the cost when w = 10 and l = 5. Inverse Variation: Eample: If y varies inversely as and y = 18 when = -, find y when = -11. The volume V of a gas kept at a constant temperature varies inversely as the pressure p. If the pressure is 4 pounds per square inch, the volume is 15 cubic feet. What will be the volume when the pressure is 0 pounds per square inch? 14

15 Variation Homework: 6. Suppose y varies jointly with and z. If y z = 0 when and 5, = = find y when z = = 14 and 8. Write an equation of variation to represent the situation and solve for the indicated information. 7. The value of real estate V varies jointly with the neighborhood inde N and the square footage of the house S. If V = $76, 0 when N = 96 and S = 1600, find the value of a property with N = 8 and S = The number of gallons g of fuel used on a trip varies directly with the number of miles m traveled. If a trip of 70 miles required 1 gallons of fuel, how many gallons are required for a trip of 400 miles? 9. The time to complete a project varies inversely with the number of employees. If people can complete the project in 7 days, how long will it take 5 people? 10. A bottle of 150 vitamins costs $5.5. If the cost varies directly with the number of vitamins in the bottle, what should a bottle with 50 vitamins cost? 15

16 11. The number of revolutions made by a tire traveling over a fied distance varies inversely to the radius of the tire. A 1-inch radius tire makes 100 revolutions to travel a certain distance. How many revolutions would a 16-inch radius tire require to travel the same distance? 1. Wei received $55.5 in interest on the $10 in her credit union account. If the interest varies directly with the amount deposited, how much would Wei receive for the same amount of time if she had $000 in the account? 1. Karen earns $8.50 for working si hours. If the amount m she earns varies directly with h the number of hours she works, how much will she earn for working 10 hours? 14. The number of gallons g in a circular swimming pool varies jointly with the square of the radius r and the depth d. If g = 754 when r = 4 and d =, find the number of gallons in the pool when r = and d = The time needed to travel a certain distance varies inversely with the rate of speed. If it takes 8 hours to travel a certain distance at 6 miles per hour, how long will it take to travel the same distance at 60 miles per hour? 16

17 17 To Solve a Rational Equation: Eamples: = 4. p p p p p p = = 10 = = = n n n n n

18 Solve Rational Equations Homework: Solve. Check for etraneous solutions =. = 1. = 5 4( 4) 4 4. = = = = = 18

19 = =

20 WORK PROBLEMS: Word Problems Eamples: Bill s garden hose can fill the pool in 1 hours. His neighbor has a hose that can fill the pool in 15 hours. How long will it take to fill the pool using both hoses? Joe can complete his yard work in hours. If his son helps, it will only take hours working together. How long would the yard work take if his son works alone? If a certain number is subtracted from the numerator and added to the denominator of the new fraction is equivalent to Find the number. In the United States, 7 out of every 5 homes are heated by electricity. At this rate, how many homes in a community of 6,000 homes would you predict are heated by electricity? 0

21 In the United States 1 out of 50 homes is heated by wood. At this rate, how many homes in a community of 6,000 homes are heated by wood? Melissa can clean the house in 4 hours, whereas her husband Zack, can do the same job in 5 hours. They have agreed to clean together so that they can finish in time to watch a movie that starts in hours. How long will it take them to clean the house together? Will they finish before the movie starts? Steve takes 1.5 times as long to go 7 miles upstream in his boat as he does to return. If the boat cruises at 0 mph in still water, what is the speed of the current? A fisherman traveling on the river takes / times as long to travel 60 miles downstream in his boat than to return. If the boat s speed is 5 mph in still water, find the speed of the current. 1

22 Word Problem Homework: Solve each equation The average cost of producing graphing calculators is given by the function f ( ) = Find the number of graphing calculators that must be produced for the average cost to be $ The average cost of producing boes of cereal is given by the function f ( ) = Find the number of boes of cereal that must be produced for the average cost to be $.00.. An elevated water reservoir supplies water to two nearby towns. One town can empty the reservoir in 10 hours by itself. The other town can empty the reservoir in 15 hours by itself. It takes 0 hours for an inlet pip to completely fill the reservoir. If the reservoir is completely filled and the two outlet for the towns are opened at the same time as the inlet pipe for filling the reservoir how long will it take to completely drain the reservoir? 4. What number must be subtracted from both the numerator and denominator of 19 7 so the resulting fraction is equivalent to 4? 5. The difference of a number and 1 times its reciprocal is 4. Find the number(s). 6. The quotient of a number and twice its reciprocal is 8. Find the number(s). 7. The quotient of a number and times its reciprocal is 4. Find the number(s).

23 8. The speed of a current is 6 mph. If it takes a boat the same time to travel 8 miles upstream as it takes to travel 16 miles downstream, find the speed of the boat in still water. 9. The speed of a person on in-line rollers skates is 4 mph slower than the speed of a bicyclist. If the in-line skater can ravel 5 miles in the same amount of time the bicyclist can travel 8 miles, find the speed of the in-line skater. 10. Mr. Kelly can paint a fence around his house in 6 hours. His son needs 1 hours to do the same job. How many hours would it take to do the job if they worked together? 11. One person can rake a yard in 4 hours. If a second person helps with the raking, it takes only hours to rake the yard. How long would it take the second person to rake the yard working alone? 1. Normally, an above-ground swimming pool can be filled with a hose in 6 hours. The pool developed a lean that drained the pool in 6 hours. How long will it take to fill the pool if the leak is not repaired first? 1. A faucet can fill a tub in 7 minutes. It takes twice that long for the drain to empty the tub. How long will it take to fill the tub if the faucet is on and drain is open? 14. Your wages vary directly as the number of hours you work. You can earn $440 when you work 44 hours. How much would you earn if you worked 50 hours? 15. The time that it takes you to drive to work varies inversely as your driving rate. If you average 5 miles per hour, it takes you minutes to get to work. How long would it take you to get to work if you average speed was 45 miles per hour? 16. The number of hours it takes a math professor to grade final eams varies jointly as the number of students in each class and the number of class the professor teaches. If it takes the professor 7 hours to grade final eams when there are students in each of the classes the professor teaches, how long will it take to grade final eams when the professor teaches 4 classes of 1 students each?

24 17. At an amusement park, the daily sales of ice cream are directly proportional to the square of the daily high temperature. IF 698 ice cream cones are sold on a day when the high temperature is 86 0 Fahrenheit, how many ice cream cones will be sold when it is 75 0 F? 18. The voltage V in an electric circuit is held constant, the current I is inversely proportional to the resistance R. If the current is amperes when the resistance is 5 ohms, find the current when the resistance is 0 ohms. 19. The area of an ellipse varies jointly as the length of its major ais a and the length of its minor ais b. If the area of an ellipse with a =8 cm and b = 18 cm is 16π square centimeters, what is the area of an ellipse with a = 54 cm and b = 16 cm? 0. If one person can do a job in 10 hours and a second person can do the same job in 14 hours. How long will it take them to do the job working together? 1. Andi can wa a car three times faster than Cindy. If they work together, they can wa the car in one hour. How long will it take each one to wa the car alone?. Suppose that R is inversely proportional to w. If R = 10 and w = 8, find w when R = 00.. Suppose that P is jointly proportional to t and s. If P = 60 when t = 5 and s =, find P when t = 0 and s = Hooke s Law states that the distance d that an elastic object such as a spring is stretched by placing a certain weight on it varies directly as the weight w of the object. If the distance is 40 cm when the weight is kg. what is the distance when a -kg weight is attached? 5. The amount of garbage G produced in the United States varies directly with the number of people N who produce the garbage. It is know that 50 tons of garbage is produced by 00 people in 1 year. The population of San Francisco is 705,000. How much garbage is produced by the people of San Francisco in 1 year? 4

25 6. Crane A can unload a container ship in 10 hours, and Crane B can unload it in 14 hours. Crane A started to unload the ship at noon and was joined by Crane B at P.M. At what time was the unloading job of the ship complete? 7. A nurse wishes to obtain 800 ml of a 7% solution of boric acid by miing 4% and 1% solutions. How much of each should be used? 8. Pump A can drain the swimming pool in 0 hours and pump B can drain it in 4 hours. Because of the quick change in weather both pumps were used to drain the pool so proper cleaning could take place for winter. How long did it take for the pumps to drain the pool? 9. How much pure antifreeze must be added to 1 L of a 40% solution of antifreeze to obtain a 60% solution? 0. An old conveyor belt takes 1 hours to move one day s coal output from the mine to a rail line. A new belt can do it in 15 hours. How long does it take when both are used at the same time? 1. Pam jogged up a hill at 6km/hr and jogged back down at 10km/hr. How many kilometers did she travel in all if her total jogging time was 1 hour and 0 minutes?. A commercial jet can fly from San Francisco to Dallas in hours. A private jet can make the same trip in ½ hours. If the two planes leave San Francisco at noon, after how many hours dis the private jet twice as far from Dallas as the commercial jet?. If inlet pipes can fill a pool in one hour and 0 minutes, and one pipe can fill the pool in two hours and 0 minutes on its own, how long would it take the other pipe to fill the pool on its own? 4. It takes 6 minutes less time to travel 10 miles by car at night than by day because the lack of traffic allows the average speed at night to be 10 miles per hour faster than in the daytime. Find the average speed in the daytime. 5

26 5. An investor bought a number of shares of stock for $1,600. After the price dropped by $10 per share, the investor sold all but 4 of her shares for $1,10. How many shares did she originally by? 6. Suppose that =. Show that when = and =, the value of t does not depend on the value of a. For which values of a do these relationships have no meaning? 7. Consider an ecosystem of rabbits in a park that starts with 10 rabbits and can sustain up to 60 rabbits. An equation that roughly models this scenario is = where P represents the rabbit population in year t of the study. a. What is the rabbit population in year 10? Round your answer to the nearest whole rabbit. b. Solve this equation for t. describe what this equation represents in the contet of this problem c. At what time does the population reach 50 rabbits? 6

27 Dividing Polynomials Remember Long Division? Dividing by a monomial- 9 y 15y y 1y 5a b 15ab 10a b 5ab 4 Dividing by a polynomial ( 4 )( ) 1 7

28 Dividing Polynomials Homework Day 1: 8

29 Dividing with Synthetic Division Synthetic Division Step 1: Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients. Step : Step : Step 4: Step 5: Eamples:. Use synthetic division to find ( 4 6 4) ( ) by 5 4 by 9

30 Dividing Polynomials Homework Day : 0