Applications of Rational Expressions

Transcription

1 6.5 Applications of Rational Expressions 1. Find the value of an unknown variable in a formula. 2. Solve a formula for a specified variable. 3. Solve applications using proportions. 4. Solve applications about distance, rate, and time. 5. Solve applications about work rates. Copyright 2016, 2012, 2008 Pearson Education, Inc. 1

2 Objective 1 Find the value of an unknown variable in a formula. Copyright 2016, 2012, 2008 Pearson Education, Inc. 2

3 Example 1 Finding the Value of a Variable in a Formula Use the formula = + to find p if f = 15 cm and q = 25 cm. f p q = + f p q = + 15 p p p = + p 25 5 p = p 2 p = p = 2 Copyright 2016, 2012, 2008 Pearson Education, Inc. 3

4 Objective 2 Solve a formula for a specified variable. Copyright 2016, 2012, 2008 Pearson Education, Inc. 4

5 Example 2 Solve for q = p q r Solving a Formula for a Specified Variable pqr + = pqr p q r 3qr + 3pr = 5pq 3pr = 5pq 3qr 3 pr = q(5 p 3 r) 3pr 5p 3r = q Copyright 2016, 2012, 2008 Pearson Education, Inc. 5

6 Example 3 Solve for R. Solving a Formula for a Specified Variable A = Rr R + r Rr ( R+ r) A= ( R+ r) R + r AR+ Ar = Rr AR Rr = Ar R( A r) = Ar R = Ar Ar or A r r A Copyright 2016, 2012, 2008 Pearson Education, Inc. 6

7 Objective 3 Solve applications using proportions. Copyright 2016, 2012, 2008 Pearson Education, Inc. 7

8 Example 4 Solving a Proportion In a recent year, approximately 9.8% (that is, 9.8 of every 100) of the 74,165,000 children under 18 yr of age in the United States had no health insurance. How many such children were uninsured? (Source: U.S. Census Bureau.) Let x = the number who had no health insurance. 9.8 x = ,165, x 74,165,000 = 74,165, ,165, , = x 7,268,170 = x Copyright 2016, 2012, 2008 Pearson Education, Inc. 8

9 Example 5 Solving a Proportion Involving Rates Lauren s car uses 15 gal of gasoline to drive 390 mi. She has 6 gal of gasoline in the car, and she wants to know how much more gasoline she will need to drive 800 mi. If we assume that the car continues to use gasoline at the same rate, how many more gallons will she need? Let x = the additional number of gallons needed. She knows that she can drive 390 mi with 15 gal of gasoline. She wants to drive 800 mi using (6 + x) gallons of gasoline. Set up a proportion = x Copyright 2016, 2012, 2008 Pearson Education, Inc. 9

10 Example 5 Solving a Proportion Involving Rates (cont.) = x = x 26(6 + x) = x = x = 644 x 24.8 She will need about 24.8 more gallons of gasoline. Copyright 2016, 2012, 2008 Pearson Education, Inc. 10

11 Objective 4 Solve applications about distance, rate, and time. Copyright 2016, 2012, 2008 Pearson Education, Inc. 11

12 Example 6 Solving a Distance, Rate, Time Problem A plane travels 100 mi against the wind in the same time that it takes to travel 120 mi with the wind. The wind speed is 20 mph. Find the rate of the plane in still air. Step 1 Read the problem. We must find the rate of the plane in still air. Step 2 Assign a variable. Let x = the rate of the plane in still air. Complete the table on the next slide using d = rt. Copyright 2016, 2012, 2008 Pearson Education, Inc. 12

13 Example 6 Solving a Distance, Rate, Time Problem (cont.) Against Wind With Wind d r t 100 x x x x + 20 Step 3 Write an equation. Since the time against the wind equals the time with the wind, we set up this equation = x 20 x+ 20 Copyright 2016, 2012, 2008 Pearson Education, Inc. 13

14 Example 6 Solving a Distance, Rate, Time Problem (cont.) Step 4 Solve the equation = x 20 x ( x 20)( x+ 20) = ( x 20)( x+ 20) x 20 x ( x+ 20) = 120( x 20) 100x = 120x = 20x 220 = x Step 5 State the answer. The rate of the airplane is 220 mph in still air. Copyright 2016, 2012, 2008 Pearson Education, Inc. 14

15 Example 7 Solving a Distance, Rate, Time Problem Dona drove 300 mi north from San Antonio, mostly on the freeway. She usually averaged 55 mph, but an accident slowed her speed through Dallas to 15 mph. If her trip took 6 hr, how many miles did she drive at the reduced rate? Step 1 Read the problem. We must find how many miles she drove at the reduced speed. Step 2 Assign a variable. Let x = the distance at the reduced speed. Use d = rt to complete the table (next slide). Copyright 2016, 2012, 2008 Pearson Education, Inc. 15

16 Example 7 Solving a Distance, Rate, Time Problem (cont.) Normal Speed Reduced Speed d r t 300 x x 55 x 15 x 15 Steps 3/4 Write an equation. Time on + Time at = 6 freeway reduced speed 300 x x + = Copyright 2016, 2012, 2008 Pearson Education, Inc. 16

17 Example 7 Step 5 State the answer. She drove Solving a Distance, Rate, Time Problem (cont.) 300 x x + = x x = 165(6) (300 x) + 11x= x+ 11x= 990 8x = 90 x = 90 1 or mi at the reduced speed. Copyright 2016, 2012, 2008 Pearson Education, Inc. 17

18 Objective 5 Solve applications about work rates. Copyright 2016, 2012, 2008 Pearson Education, Inc. 18

19 Rate of Work If a job can be accomplished in t units of time, then the rate of work is 1 job per unit of time. t Copyright 2016, 2012, 2008 Pearson Education, Inc. 19

20 Example 8 Solving a Work Problem Stan needs 45 min to do the dishes, while Bobbie can do them in 30 min. How long will it take them if they work together? Step 1 Read the problem. We must find how many minutes it will take them to do the dishes if they work together. Step 2 Assign a variable. Let x = the time it will take working together. Complete the table on the next slide. Copyright 2016, 2012, 2008 Pearson Education, Inc. 20

21 Example 8 Table Stan Bobbie Rate Solving a Work Problem (cont.) Time Working Together Part by Stan Part by Bobbie x + 30 x = x x Fractional Part Done 1 45 x 1 30 x 1 whole 1 Copyright 2016, 2012, 2008 Pearson Education, Inc. 21

22 Example 8 Solve. The LCD is x+ x= Solving a Work Problem (cont.) x+ x = x+ 3x= 90 5x = 90 x =18 It will take them 18 min working together. Copyright 2016, 2012, 2008 Pearson Education, Inc. 22