Answers to All Exercises

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1 Answers to All Eercises CHAPTER 5 CHAPTER 5 CHAPTER 5 CHAPTER REFRESHING YOUR SKILLS FOR CHAPTER 5 1a. between 3 and 4 (about 3.3) 1b. between 6 and 7 (about 6.9) 1c. between 7 and 8 (about 7.4) 1d. between 8 and 9 (about 8.) a. b. c. d a. 6 b c. 3 5 d. 10 e a. iii, B 3b. i, C 3c. iv, D 3d. ii, A 4a. 4b. 4c. 4d. Answers to All Eercises ANSWERS TO ALL EXERCISES 51

2 Answers to All Eercises LESSON 5.1 1a. f (5) 3.5 1b. g(14) 19,58.3 1c. h(4).9 1d. j(37) a. 16, 1, 9; 16(0.75) b. 4, 36, 54; 4(1.5) 3a. f(0) 15, f(1) 75, f() 45; u 0 15 and u n 0.6u n 1 where n 1 3b. f(0) 3, f(1) 6, f() 1; u 0 3 and u n u n 1 where n 1 4a. 0.75; 5% decrease 4b. 1. _ 3 ; 33. _ 3 % increase 4c. 0.94; 6% decrease 4d ; 6.38% increase 5a. u , u n u n b. Year Estimated population (billions) c. represents the es timated population ears after 1995; 1.11 (1.015). 5d. The equation predicts that the population of China in 006 was 1.46 billion. This is larger than the actual value. This means that the population is growing at a slower rate than it was in a. Let represent the number of the da, and let represent the height in cm..56(.5). For the fifth da,.56(.5) 5 50 cm; for the sith da,.56 (.5) 6 65 cm. 6b..56(.5) c. 78 cm 6d da, or 18 hours 6e. 11 das 13 hours, or 9 P.M. on da 11 7a d. 7e. As the base increases, the graph becomes steeper. The curves all intersect the -ais at (0, 1). 7f. The graph of 6 should be the steepest of all of these. It will contain the points (0, 1) and (1, 6). 8a. 3 8b., or 8c. 3, or 3 8d. /3 9a d. 9e. As the base increases, the graph flattens out. The curves all intersect the -ais at (0, 1). 9f. The graph of 0.1 should be the steepest of all of these. It will contain the points (0, 1) and ( 1, 10). 10a , or b. ( 5) 0.5, or c , d /, 3(0.5) / 11a b. f () 30(0.9) 30 11c. 30 g() f() d. g(4) 30 11e. possible answer: g() 30(0.9) 4 5 ANSWERS TO ALL EXERCISES

3 11f. Sample answer: You can use the - and -values of an point on the curve and the common ratio to write the equation. 1. Answers will var but will be in the form , with 1, 1 being an point from the table. 13a. Let represent time in seconds, and let represent distance in meters. 14b ( 3), ( 6), or a. A 5000( ) 5 15b. S 500(1 0.03) 5 15c. After 14 ears, Austin will have $8,093.47, and Sami will have $8,08. 16a (0, 10) (7, 10) (3.5, 3) 5 13b. domain: 0 7; range: c a. f (3) b. 4 16c. es 16d. A rectangle diagram also uses the distributive propert. Each term in the first binomial is multiplied b each term in the second binomial. Answers to All Eercises ANSWERS TO ALL EXERCISES 53

4 Answers to All Eercises 1a d LESSON 5. a. a 5 b. b 4 1b. 36 1c e f. 9 c. c 0 d. e 3 3a. False. Valid reasons include: You must have the same base for the product propert of eponents; ,831,808. 3b. False. You must raise to the power before multipling. 3c. False. Valid reasons include: You must raise to the power before dividing; onl one factor of 4 can be divided out; or it should be d. true 4a. 4b. 3 4c. 5 4d. 0 5a b c d e f. 1 6a. 1 6b c d e f Sample answer: (a b) n is not necessaril equivalent to a n b n. For eample, ( 3 ) 5 but However, the are equivalent when n 1, or when a b 0 and n is odd. 8a. 49; ; ; ; 343 8b ; ; ; The sequence is not arithmetic because there is not a common difference. 8c. 1.67; 1.67; 1.67; The ratio of consecutive terms is alwas the same, so the sequence is growing eponentiall. 8d. Possible answer: Non-integer powers ma produce non-integer values. If the eponents form an arithmetic sequence, the decimal powers form a geometric sequence. 9a d. 9e. Sample answer: As the eponents increase, the graphs get narrower horizontall or steeper verticall. The even-power functions are U-shaped and alwas in the first and second quadrants, whereas the odd-power functions have onl the right half of the U, with the left half pointed down in the third quadrant. The all pass through (0, 0) and (1, 1). 9f. Sample answer: The graph of 6 will be U-shaped, will be narrower (or steeper) than 4, and will pass through (0, 0), (1, 1), ( 1, 1), (, 64), and (, 64). Sample answer: The graph of 7 will fall in the first and third quadrants, will be narrower (or steeper) than 3 or 5, and will pass through (0, 0), (1, 1), ( 1, 1), (, 18), and (, 18). 9g. Power functions go through the origin and have long-run values of infinit. Eponential functions have -intercepts at 1 (or a) and go to 0, either as increases or as decreases. 10a. 4 3, or b. ( ) 3 10c. 4 3, or d. 8( ) 3, or a. 47(0.9)(0.9) 1 47(0.9 ) 1 (0.9) 1 47(0.9 ) b the product propert of eponents; 4.3(0.9 ) 1. 11b (0.9 ) 8 3, or 1 11c. The coefficients are equal to the values of f 1 corresponding to the number subtracted from in the eponent. If ( 1, 1 ) is on the curve, then an equation 1 b ( 1 ) is an eponential equation for the curve. 1a r 3 ; 5. r 6 1b r 3 5. r 6 ; r c. 173 cm 1 13a. 7 13b. 13c. 0 14a b. 4( ) 00 14c. 39.8( ) ANSWERS TO ALL EXERCISES

5 14d. 4( ) ; 39.8 (0.9476) ; both equations give approimatel 137. rads. 14e. 4( ) ; 39.8( ) ; both equations give 7.3 rads. 14f. 4( ) 00 4(0.9476) (0.9476) ( ) a. 7 15b. 4 15c. 4 15d ( 4 ) 3 17a. Let represent time in seconds, and let represent distance in meters b. All ou need is the slope of the median-median line, which is determined b M 1 (8, 1.6) and M 3 (31, 6.). The slope is 0.. The speed is approimatel 0. m/s Answers to All Eercises ANSWERS TO ALL EXERCISES 55

6 Answers to All Eercises LESSON a e j; b d g; c i; f h a. Power; the base is a variable. b. Power; the base is a variable. c. Eponential; the eponent is a variable. d. Power; is equivalent to 1. e. Power; a square root is equivalent to the eponent 1_. f. Power; t 4t 3 is equivalent to (t ) 1. g. Eponential; 1 is equivalent to 1(3 ) 3 t. t h. Power; 8 w 5 is equivalent to 8(w 5 ) 1. i. Power; 8 is equivalent to j. Neither; the function is not a transformation of either a or b. k. Power; the fifth root of a cube is equivalent to the eponent 3_ 5. l. Eponential; the eponent contains a variable. 3a. a 1/6 3b. b 4/5, b 8/10, or b 0.8 3c. c 1/, or c 0.5 3d. d 7/5, or d 1.4 4a. a 1/6 4.; raise both sides to the power of 6: a b. b 4/5 14.3; raise both sides to the power of 5_ 4 : b / c. c 1/ 0.55; raise both sides to the power of : c d. d 7/5 3; raise both sides to the power of 5_ 7 : d 3 5/ W/c m 6a d. 6e. Each curve is less steep than the prior one. The graphs of 1/ and 1/4 are in onl the first quadrant, whereas the graphs of 1/3 and 1/5 are in the first and third quadrants. All of the functions go through (0, 0) and (1, 1). The graphs of 1/3 and 1/5 both go through ( 1, 1). 6f. 1/7 will be less steep than the others graphed and will be in the first and third quad rants. It will pass through (0, 0), (1, 1), and ( 1, 1). 6g. The domains of 1/ and 1/4 are 0 because ou can t take a square root or fourth root of a negative num ber. The domains of 1/3 and 1/5 are all real numbers. 7a d. 7e. Each graph is steeper and less curved than the previous one. All of the functions go through (0, 0) and (1, 1). 4/4 (or ) is not curved at all. 7f. 5 4 should be steeper and should curve upward. 8. Sample answer: Power func tions with rational eponents can have limited domain. When the eponent is between 0 and 1, the curve increases slowl with a shape similar to. Eponential curves alwas have a steadil increasing or decreasing slope, unlike power functions. 9a. eponential 9b. neither 9c. eponential 9d. power 10a. 3 ( ) 3/4 10b. 1 [ ( 5) ] 3/4 10c. 4 10d /4 11a /4, or 4 3 3/4 56 ANSWERS TO ALL EXERCISES

7 11b / c. 35 3/ a AU 1b r 1c. Planet Mercur Venus Earth Mars Orbital radius (AU) Orbital time (r) Planet Jupiter Saturn Uranus Neptune Orbital radius (AU) Orbital time (r) ; PV k V 13b. k (40)(1.3) 49 13a. P k V 1 ; P k 13c. 8. L 13d. 3.8 mm Hg 14a b c d e a. ( 4 ) 15b. 1 15c. ( 5 ) 15d. ( 3 ) 4 15e. 3 15f. 1 15g. 1 15h about a. u 1 0 and u n 1. u n 1 where n 17b. u 9 86; about 86 rat sightings 17c. Let represent the ear number, and let represent the number of rats; 0(1. ) 1. Answers to All Eercises ANSWERS TO ALL EXERCISES 57

8 Answers to All Eercises LESSON 5.4 1a / b c. no real solution a. 65 b. 1 c. 51 d. 1( /7.8 ) e / a b c a. 100 r 6 4b. 100 r c. r 0.891; 89.1% 5a. She must replace with 7 and 1 with 1 7; 7 ( 1 7) b 1. 5b. 7 (105 7) b 1 ; /( 1) b 5c. Possible answers: 0, 00, b 0.508;, 57, b 0.510; 3, 31, b d. Possible answer: The mean of the b-values is (0.511 ) 1. 6a. 6b. Sample answer: ŷ , where is measured in units of 100,000 km. The graph of the data and the equation appear to be a good fit. 6c. approimatel 1,9,00 km 6d d 7a. 39 tons 7b. 54 ft 8a cm 8b m 9a. 1.9 g 9b. 1.8% % per month, or 3.9% per ear 11a , or 4.66% per ear 11b. 6.6 g 11c. 6.6( ) 6.6( ) 11d. 0.6 g 11e r 1a Temperature ( F) 1b. 18.9, 9.15, 40.1, 50.35, c. range 38.5, IQR 1. 1d. The data do not support his conjecture. There are approimatel the same number of cities in each categor ,, z ANSWERS TO ALL EXERCISES

9 LESSON ( 3, ), ( 1, 0), (, ), (6, 4) a. 9 b. c. 15, or Graph c is the inverse because the - and -coordinates have been switched from the original graph so that the graphs are smmetric across the line. 4. a and e are inverses; b and d are inverses; c and g are inverses; f and h are inverses. 5a. f (7) 4; g(4) 7 5b. The might be inverse functions. 5c. f (1) ; g( ) 5 5d. The are not inverse functions, at least not over their entire domains and ranges. 5e. f () for 3 and g() for 4 (its entire domain) are inverse functions. 6a. 4 ( ) 3/5 1 ( ) 3/ / b. 4 ( ) 3/5 ( ) 3/5 4 ( 4 ) 5/3 f 1 () ( 4 ) 5/3 6c. Sample answer: The steps are the same, but ou don t have to do the numerical calculations when ou find an inverse. 7a. 1, 0, 1, 7b. ( 1, 3), (0, 1), (1, 0), (, ) 7c. Yes, it is a function; it passes the vertical line test. 1, or f 1 () 3 8a. f () 3; f 1 () 3 8b. f () or f 1 () ; f 1 () 4, or f 3 1 ( ) c. f () or f 1 () ; 3 (not a function) 9a. i. f 1 ( ) a. ii. f ( f 1 (15.75)) a. iii. f 1 ( f (15.75)) a. iv. f ( f 1 ( )) f 1 ( f()) 9b. i. f 1 ( ) b. ii. f ( f 1 (15.75)) b. iii. f 1 ( f(15.75)) b. iv. f f 1 () f 1 f () 10a. The equation of the median-median line is f() b. f 1 ( ) , or f 1 () c. The equation of the median-median line is g() d. g 1 ( ) , or g 1 ( ) ,591 10e. Use the function in 10a to find the temperature in C first. f (6194) (6194) C. Then use the function from 9b to change the C to F: F. 11a. 100 C 11b. C 100 F Your friend s score is 1. Sample answers are given for eplanations of incorrect answers. Problem 1 is correct. Problem is incorrect: The notation f 1 ( ) indicates the inverse function related to f (), not the eponent 1. Problem 3 is incorrect: The epression 9 1/5 1 can be rewritten as 91/5. Problem 4 is incorrect: The epression 0 is not defined. 13a. i, ii, iii 13b. ii, v 13c. i, iv 13d. i, ii, iii 14a. c() , where c is the cost in dollars and is the number of thousands of gallons 14b. $ c. g() 7.18, where g is the number of 3.98 thousands of gallons and is the cost in dollars 14d. 1,000 gal 14e. g c() g( ) c g() c f. about $16 14g. Answers will var, but volume should equal , or 346,500 in 3 or approimatel 00 ft possible answers: 3 15, 3 15, 5, 3 15,65, f () 9 1.6(1.5 ) or f() 4.55(1.5 ) 5 17a., or b., or c ( 3 ) and 1 9 ( ) , 1, z 0 Answers to All Eercises ANSWERS TO ALL EXERCISES 59

10 Answers to All Eercises 1a b c. 7 1/ 1d e. 5 1f. 6 1 a. 3 b. 4 c d. e. 1 5 f. 0 LESSON 5.6 3a. lo g ; 3 3b. lo g 5 100; c. lo g 35 8; d. lo g 0.4 5; e. lo g ; f. lo g ; a. This is a translation of the graph of log horizontall units. Note that it actuall continues downward indefinitel. 4b. This is a vertical dilation of the graph of log b a factor of 3. 4c. This is a reflection of the graph of log across the -ais and a translation verticall units. 4d. This is a translation of the graph of 10 horizontall units. 4e. This is a vertical dilation of the graph of 10 b a factor of 3. 4f. This is a reflection of the graph of 10 across the -ais and a translation verticall units. 5a. false; log 6 1 5b. false; 5 log 5.5 5c. false; log 3 5d. false; log approimatel 5 min 7a. sometime in b. 8.3% 7c. 8.7 r 8a. 100( ) 8b. 605 r ago. The technique is approimate and assumes that the carbon-14 concentration in the atmosphere has not changed over the past 6000 r. 9a. 88.7( ) 9b. 3 or 4 clicks 10a b. 7 1/ ANSWERS TO ALL EXERCISES

11 11a. Median-median line equation using ears since 1900 is ŷ Passengers Year Passengers (millions) Years since b. residuals: 1.85, 0.15, 6.75, 17.5, 14.45, 6.5, 5.5, 19.5, 4.75, 3.95, 6.45, 51.75, 36.65, 6.75, 1.75, c Predictions based on this model will gen erall be within 5.3 million of the correct number of passengers. 11d. Based on the model, about million passengers. A better estimate might be to sa between 850 million and 900 million passengers. 1a. C 1 3.7, C 65.4, C , C , C , C b ( ), where represents C-note number and represents frequenc in ccles per second 13a. 1 3, or 4 13b. 4 ( 5 ), or ( 5 ) 4 13c. 6, or 6 13d. 7, or 7 14a. l w 155 l w 7 14b. l 54, w 3.5; length: 54 in., width: 3.5 in. 15a The are parallel. 15b. possible answer: A(0, 3); P(1, 1); Q(4, 3) 15c. Possible answer: Translate horizontall 1 unit and verticall 4 units. ( 1) 3( 4) 9. 15d. Possible answer: Translate horizontall 4 units and verticall 6 units. ( 4) 3( 6) 9. 15e. Possible answer: ( 1) 3( 4) , which is l ( 4) 3( 6) , which is also l Answers to All Eercises ANSWERS TO ALL EXERCISES 61

12 Answers to All Eercises 1a. log 55 1b. log 8 1c. log 4 1d. log e. log 63 a. log log 11 LESSON 5.7 b. man possible answers, such as log 6 log c. log 3 log 13 d. man possible answers, such as log 14 log 3a. log 5 3b. log 3c. 1 log 3 3d. log 7 4a. true 4b. false; possible answer: log 5 log 3 log 15 4c. true 4d. true 4e. false; possible answer: log 9 log 3 log 3 4f. false; possible answer: log 7 1 log 7 4g. false; possible answer: log 35 log 5 log 7 4h. true 4i. false; possible answer: log 3 log 4 log 3 4 4j. true 5a. g h g k ; product propert of eponents 5b. log st; product propert of logarithms 5c. f w v ; quotient propert of eponents 5d. log h log k; quotient propert of logarithms 5e. j st ; power propert of eponents 5f. g log b; power propert of logarithms 5g. k m/n ; definition of rational eponents 5h. log u t; change-of-base propert 5i. w t s ; product propert of eponents 1 5j. ; definition of negative eponents p h 6a. 100( ) 6b. 11,460 r 6c ; about B.C.E. 6d. 100( ) ; 100( ) 100,000,000 ; 0. There is virtuall nothing left to measure, so ou could onl use carbon-14 for dating coal if ou had ver sensitive instruments to detect the radioactivit. 7a. Let represent the note s number of steps above middle C, and let represent the note s frequenc in hertz /1 because the starting value is 61.6 and there are 1 intermediate frequencies to get to the last C note, which has double that frequenc. 7b. Note Frequenc (Hz) Do C C# 77. Re D 93.6 D# Mi E 39.6 Fa F 349. F# Sol G 39.0 G# La A A# Ti B Do C a b c d a. 14.7( ) 9b. Air pressure (lb/in. ) Altitude (mi) (0, 14.7) (, 9.467) Altitude (mi) (9.467, ) (14.7, 0) Air pressure (lb/in ) 9c lb/i n 9d. 6.3 mi 10a. 96.5% 10b. 100(0.965 ), with in minutes 10c min 10d. In one da, the carbon-11 is virtuall gone, so ou could never date an archaeological find. 6 ANSWERS TO ALL EXERCISES

13 11. Graphs will var. If a horizontal line intersects the graph in more than one point, its inverse is not a function. 1a b. 3 13a. The graph has been verticall dilated b a factor of 3, then translated horizontall 1 unit and verticall 4 units b. The graph has been horizontall dilated b a factor of 3, reflected across the -ais, and translated verticall units a. False. If everone got a grade of 86% or better, one would have to have gotten a much higher grade to be in the 86th percentile. 14b. False. Consider the data set {5, 6, 9, 10, 11}. The mean is 8.; the median is 9. 14c. False. Consider the data set {0,, 8}. The range is 8; the difference between the mean, 10, and the maimum, 8, is d. true 15a. Let h represent the length of time in hours, and let c represent the driver s cost in dollars. c 14h 0. The domain is the set of possible values of the number of hours, h 0. The range is the set of possible values of the cost paid to the driver, c 0. 15b. Let c represent the driver s cost in dollars, and let a re present the agenc s charge in dollars. a 1.15c 5. The domain is the mone paid to the driver if she had been booked directl, c 0. The range is the amount charged b the agenc, a c. a 1.15(14h 0) 5, or a 16.1h 48 Answers to All Eercises ANSWERS TO ALL EXERCISES 63

14 Answers to All Eercises 1a b. 11 1c. 4 1d e f LESSON 5.8 a. log 10 n p log 10 n 10 p (n p)log 10 log 10 n log 10 p (n p)log 10 n log 10 p log 10 (n p)log 10 (n p)log 10 Because the logarithm of the left side equals the logarithm of the right, the left and right sides are equal. Or, because log 10 n p log 10 n 10 p, 10 n p 10 n p 10. b. log 10d 10 e log 10 d e log 10 d log 10 e log 10 d e d log 10 e log 10 (d e)log 10 (d e)log 10 (d e)log 10 Because the logarithm of the left side equals the logarithm of the right, the left and right sides are equal. Or, because log 10d 10 log 10 d e, 10d e d e. log 3 e 3. t 195.9; about mo, or about log r 4 mo 4a. h 146( ) T 4 4b. about 4.1 h at 30 C; about 63.6 h at 16 C 4c ( ) T 4 ; log T 4 ; T 4 log ; T C 4d. 5d (1.09 ) 6000; 1 499(1.09 ) ; 1 499(1.09 ) ; 0.00 (1.09) ; log 0.00 log(1.09 ) ; log 0.00 log 1.09; log log e. Sample answer: The number of games sold starts out increasing slowl, then speeds up, and then slows down as everone who wants the game has purchased one a. D 10 log db 6b. D 10 log db c. I W/ cm 6d. about 3.16 times as loud 7a. 7b. (log, ) is a linear graph. 7c. 6 0; ŷ 6 0 log 7d. 4e. A realistic domain is 0 to 100 C; these are the freezing and boiling points of water. 5a. f (0) After 0 das, 133 games have been sold. 5b. f (80) After 80 das, 7969 games have been sold. 5c After 7 das, 6000 games have been sold. Sample answer: Yes; the graph shows that the equation is a good model for the data. 64 ANSWERS TO ALL EXERCISES

15 8a. Let represent time in min, and let represent temperature in F. 8b. Because the curve is both ref lected and translated, first graph points in the form (, ). Then translate the points up so that the data approach a long-run value of 0. Estimating that the new points approach the long-run value 74, graph points in the form (, log( 74)), which appears to be linear. The median-median line for these altered data is log( 74) Solving for gives the equation ŷ , or ŷ ( ). 9a. The data are the most linear when viewed as (log(height), log(distance)). 9b. The median-median line equation for these altered data is ŷ Or, in terms of the original data, log(distance) log(height). Solving for distance gives the equation ŷ log, ŷ log, ŷ log , or ŷ a qt after 1 da; qt after das; u 0 16, u n u n 1 (1 0.15) 1, n 1 10b. 11a. 18( ) 4, 144( ) 10, or 4.5( ) log log 18 11b. log 4, log log 144 log 10, or log log 4.5 log 1a. cost: ,000; income: 1.9 1b. Cost/income ($) 1c. 111,765 lb 1d. $66, ( 5, 10) 1_ ( 54, 8) 3_ ( 4 4, 8) ( 5, 6) ( 5, 8) 1,000, , ,000 50, ,000, Fish sticks (lb) 10 14a / b. 56 1/ or c / Answers to All Eercises Chlorine (qt) Das after first treatment 6. _ (0.85 ) ANSWERS TO ALL EXERCISES 65

16 1a CHAPTER 5 REVIEW 1 1b. 3 1d. 7 1e c. 15 1f g. 1 1h. 1 1i. 0.6 a. lo g 3 7 or log 3 log 7 b. log 4 5 or log 4 log 5 c. log 7 5 3a b c d. 5 11a. a b. b log 15 11c. log ; The real-world meaning of the -intercept is that the first 0.63 min of calling is free. 11d. $ e. about 4 min 1a. Answers to All Eercises 4a. log 4b. log z log v 4c log t 4d. k log w 4e. 1/5 4f. log 5 log 8 5a..153 log 4.7 5b. log log 4.7 5c.. 9 1/ d e / log 18 5f log a / b / log 734 6c log d. 0. 6e / f log ; about 39.9 h log 0.5 8a. 1 8b. ( 1 ) c. 8d ( 1)/ b. domain: 0 10; range: c. Verticall dilate b a factor of 80; reflect across the -ais; verticall shift b d. 55% 1e. about 4 r old 13a. approimatel 37 sessions 13b. approimatel 48 wpm 13c. Sample answer: It takes much longer to improve our tping speed as ou reach hig her levels. 60 wpm is a good tping speed, and ver few people tpe more than 90 wpm, so 0 90 is a reasonable domain. 14a. u 0 1, u n u n 1, n 1 14b. 14c. 14d. Answers will var but can include curving upward, increasing, increasing at an increasing rate, discrete. 14e. after 0 cell divisions 14f. after 9 divisions ANSWERS TO ALL EXERCISES