Exponential and Logarithmic Functions

Transcription

1 Exponenial and Logarihmic Funcions C MOST of he funcions we have considered so far have been polynomial or raional funcions, wih a few ohers involving roos of polynomial or raional funcions. Funcions ha can be expressed in erms of addiion, subracion, muliplicaion, division, and he aking of roos of variables and consans are called algebraic funcions. In Chaper 5 we inroduce and invesigae he properies of exponenial funcions and logarihmic funcions. These funcions are no algebraic; hey belong o he class of ranscendenal funcions. Exponenial and logarihmic funcions are used o model a variey of real-world phenomena: growh of populaions of people, animals, and baceria; radioacive decay; epidemics; absorpion of ligh as i passes hrough air, waer, or glass; magniudes of sounds and earhquakes. We consider applicaions in hese areas plus many more in he secions ha follow. 5 CHAPTER SECTIONS 5-1 Exponenial Funcions 5-2 Exponenial Models 5-3 Logarihmic Funcions 5-4 Logarihmic Models 5-5 Exponenial and Logarihmic Equaions Chaper 5 Review Chaper 5 Group Aciviy: Comparing Regression Models Cumulaive Review Chapers 4 and 5

2 470 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 5-2 Exponenial Models Z Mahemaical Modeling Z Daa Analysis and Regression Z A Comparison of Exponenial Growh Phenomena In Secion 5-2 we use exponenial funcions o model a wide variey of real-world phenomena, including growh of populaions of people, animals, and baceria; radioacive decay; spread of epidemics; propagaion of rumors; ligh inensiy; amospheric pressure; and elecric circuis. The regression echniques inroduced in Chapers 2 and 3 o consruc linear and quadraic models are exended o consruc exponenial models. Z Mahemaical Modeling Populaions end o grow exponenially and a differen raes. A convenien and easily undersood measure of growh rae is he doubling ime ha is, he ime i akes for a populaion o double. Over shor periods he doubling ime growh model is ofen used o model populaion growh: P P 0 2 d where P Populaion a ime P 0 Populaion a ime 0 d Doubling ime Noe ha when d, P P 0 2 dd P 0 2 and he populaion is double he original, as i should be. We use his model o solve a populaion growh problem in Example 1. EXAMPLE 1 Populaion Growh Nicaragua has a populaion of approximaely 6 million and i is esimaed ha he populaion will double in 36 years. If populaion growh coninues a he same rae, wha will be he populaion: (A) 15 years from now? (B) 40 years from now?

3 S E C T I O N 5 2 Exponenial Models 471 SOLUTIONS We use he doubling ime growh model: P P 0 2 d Subsiuing P 0 6 and d 36, we obain P 6(2 36 ) Figure Years 50 Z Figure 1 P 6(2 36 ). (A) Find P when 15 years: (B) Find P when 40 years: P 6( ) 8 million P 6( ) 13 million MATCHED PROBLEM 1 The bacerium Escherichia coli (E. coli) is found naurally in he inesines of many mammals. In a paricular laboraory experimen, he doubling ime for E. coli is found o be 25 minues. If he experimen sars wih a populaion of 1,000 E. coli and here is no change in he doubling ime, how many baceria will be presen: (A) In 10 minues? (B) In 5 hours? Wrie answers o hree significan digis.

4 472 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS ZZZ EXPLORE-DISCUSS 1 The doubling ime growh model would no be expeced o give accurae resuls over long periods. According o he doubling ime growh model of Example 1, wha was he populaion of Nicaragua 500 years ago when i was seled as a Spanish colony? Wha will he populaion of Nicaragua be 200 years from now? Explain why hese resuls are unrealisic. Discuss facors ha affec human populaions ha are no aken ino accoun by he doubling ime growh model. As an alernaive o he doubling ime growh model, we can use he equaion y ce k where y Populaion a ime c Populaion a ime 0 k Relaive growh rae The relaive growh rae k has he following inerpreaion: Suppose ha y ce k models he populaion growh of a counry, where y is he number of persons and is ime in years. If he relaive growh rae is k 0.03, hen a any ime, he populaion is growing a a rae of 0.03y persons (ha is, 3% of he populaion) per year. Example 2 illusraes his approach. EXAMPLE 2 Medicine Baceria Growh Cholera, an inesinal disease, is caused by a cholera bacerium ha muliplies exponenially by cell division as modeled by N N 0 e where N is he number of baceria presen afer hours and N 0 is he number of baceria presen a 0. If we sar wih 1 bacerium, how many baceria will be presen in (A) 5 hours? (B) 12 hours? Compue he answers o hree significan digis. SOLUTIONS (A) Use N 0 1 and 5: N N 0 e Le N 0 1 and 5. e 1.386(5) 1,020 Calculae o hree significan digis.

5 S E C T I O N 5 2 Exponenial Models 473 (B) Use N 0 1 and 12: N N 0 e Le N 0 1 and 12. e 1.386(12) 16,700,000 Calculae o hree significan digis. MATCHED PROBLEM 2 Repea Example 2 if N N 0 e and all oher informaion remains he same. Exponenial funcions can also be used o model radioacive decay, which is someimes referred o as negaive growh. Radioacive maerials are used exensively in medical diagnosis and herapy, as power sources in saellies, and as power sources in many counries. If we sar wih an amoun A 0 of a paricular radioacive isoope, he amoun declines exponenially in ime. The rae of decay varies from isoope o isoope. A convenien and easily undersood measure of he rae of decay is he half-life of he isoope ha is, he ime i akes for half of a paricular maerial o decay. We use he following half-life decay model: A A 0 ( 1 2) h A 0 2 h where A Amoun a ime A 0 Amoun a ime 0 h Half-life Noe ha when h, A A 0 2 h h A A 0 2 and he amoun of isoope is half he original amoun, as i should be. EXAMPLE 3 Radioacive Decay The radioacive isoope gallium 67 ( 67 Ga), used in he diagnosis of malignan umors, has a biological half-life of 46.5 hours. If we sar wih 100 milligrams of he isoope, how many milligrams will be lef afer (A) 24 hours? (B) 1 week? Compue answers o hree significan digis.

6 474 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS SOLUTIONS We use he half-life decay model: A A 0 ( 1 2) h A 0 2 h Using A and h 46.5, we obain A 100( ) Figure A (milligrams) Hours Z Figure 2 A 100( ). 200 (A) Find A when 24 hours: A 100(2 24/46.5 ) 69.9 milligrams Calculae o hree significan digis. (B) Find A when 168 hours (1 week 168 hours): A 100(2 168/46.5 ) 8.17 milligrams Calculae o hree significan digis. MATCHED PROBLEM 3 Radioacive gold 198 ( 198 Au), used in imaging he srucure of he liver, has a halflife of 2.67 days. If we sar wih 50 milligrams of he isoope, how many milligrams will be lef afer: 1 2 (A) day? (B) 1 week? Compue answers o hree significan digis. As an alernaive o he half-life decay model, we can use he equaion y ce k, where c and k are posiive consans, o model radioacive decay. Example 4 illusraes his approach.

7 S E C T I O N 5 2 Exponenial Models 475 EXAMPLE 4 Carbon-14 Daing Cosmic-ray bombardmen of he amosphere produces neurons, which in urn reac wih nirogen o produce radioacive carbon-14. Radioacive carbon-14 eners all living issues hrough carbon dioxide, which is firs absorbed by plans. As long as a plan or animal is alive, carbon-14 is mainained in he living organism a a consan level. Once he organism dies, however, carbon-14 decays according o he equaion A A 0 e where A is he amoun of carbon-14 presen afer years and A 0 is he amoun presen a ime 0. If 1,000 milligrams of carbon-14 are presen a he sar, how many milligrams will be presen in (A) 10,000 years? (B) 50,000 years? Compue answers o hree significan digis. SOLUTIONS Subsiuing A 0 1,000 in he decay equaion, we have A 1,000e Figure 3 A 1, ,000 Z Figure 3 (A) Solve for A when 10,000: A 1,000e (10,000) 289 milligrams Calculae o hree significan digis. (B) Solve for A when 50,000: A 1,000e (50,000) 2.03 milligrams Calculae o hree significan digis. More will be said abou carbon-14 daing in Exercise 5-5, where we will be ineresed in solving for afer being given informaion abou A and A 0.

8 476 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS MATCHED PROBLEM 4 Referring o Example 4, how many milligrams of carbon-14 would have o be presen a he beginning o have 10 milligrams presen afer 20,000 years? Compue he answer o four significan digis. We can model phenomena such as learning curves, for which growh has an upper bound, by he equaion y c(1 e k ), where c and k are posiive consans. Example 5 illusraes such limied growh. EXAMPLE 5 Learning Curve People assigned o assemble circui boards for a compuer manufacuring company undergo on-he-job raining. From pas experience, i was found ha he learning curve for he average employee is given by N 40(1 e 0.12 ) where N is he number of boards assembled per day afer days of raining (Fig. 4). N Days 50 Z Figure 4 N 40(1 e 0.12 ). (A) How many boards can an average employee produce afer 3 days of raining? Afer 5 days of raining? Round answers o he neares ineger. (B) Does N approach a limiing value as increases wihou bound? Explain.

9 S E C T I O N 5 2 Exponenial Models 477 SOLUTION (A) When 3, N 40(1 e 0.12(3) ) 12 Rounded o neares ineger so he average employee can produce 12 boards afer 3 days of raining. Similarly, when 5, N 40(1 e 0.12(5) ) 18 Rounded o neares ineger Because e 0.12 approaches 0 as increases wihou bound, N 40(1 e 0.12 ) S 40(1 0) 40 So he limiing value of N is 40 boards per day. (Noe he horizonal asympoe wih equaion N 40 ha is indicaed by he dashed line in Fig. 4.) MATCHED PROBLEM 5 A company is rying o expose as many people as possible o a new produc hrough elevision adverising in a large meropolian area wih 2 million poenial viewers. A model for he number of people N, in millions, who are aware of he produc afer days of adverising was found o be N 2(1 e ) (A) How many viewers are aware of he produc afer 2 days? Afer 10 days? Express answers as inegers, rounded o hree significan digis. (B) Does N approach a limiing value as increases wihou bound? Explain. We can model phenomena such as he spread of an epidemic or he propagaion of a rumor by he logisic equaion. M y (1 ce k ) where M, c, and k are posiive consans. Logisic growh, illusraed in Example 6, approaches a limiing value as increases wihou bound. EXAMPLE 6 Logisic Growh in an Epidemic A communiy of 1,000 individuals is assumed o be homogeneously mixed. One individual who has jus reurned from anoher communiy has influenza. Assume he communiy has no had influenza shos and all are suscepible. The spread of he disease in he communiy is prediced o be given by he logisic curve 1,000 N() 1 999e 0.3 where N is he number of people who have conraced influenza afer days (Fig. 5).

10 478 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS N 1,500 1, Days 50 Z Figure 5 1,000 N 1 999e 0.3. (A) How many people have conraced influenza afer 10 days? Afer 20 days? Round answers o he neares ineger? (B) Does N approach a limiing value as increases wihou bound? Explain. SOLUTIONS (A) When 10, 1,000 N (10) 1 999e Rounded o neares ineger so 20 people have conraced influenza afer 10 days. Similarly, when 20, 1,000 N (20) 1 999e Rounded o neares ineger so 288 people have conraced influenza afer 20 days. (B) Because e 0.3 approaches 0 as increases wihou bound, N 1,000 1,000 S 1, e (0) So he limiing value is 1,000 individuals (all in he communiy will evenually conrac influenza). (Noe he horizonal asympoe wih equaion N 1,000 ha is indicaed by he dashed line in Fig. 5.) MATCHED PROBLEM 6 A group of 400 parens, relaives, and friends are waiing anxiously a Kennedy Airpor for a charer fligh reurning sudens afer a year in Europe. I is sormy and he plane is lae. A paricular paren hough he had heard ha he plane s radio had

11 S E C T I O N 5 2 Exponenial Models 479 gone ou and relaed his news o some friends, who in urn passed i on o ohers. The propagaion of his rumor is prediced o be given by 400 N() 1 399e 0.4 where N is he number of people who have heard he rumor afer minues. (A) How many people have heard he rumor afer 10 minues? Afer 20 minues? Round answers o he neares ineger. (B) Does N approach a limiing value as increases wihou bound? Explain. Z Daa Analysis and Regression We use exponenial regression o fi a funcion of he form y ab x o a se of daa poins, and logisic regression o fi a funcion of he form c y 1 ae bx o a se of daa poins. The echniques are similar o hose inroduced in Chapers 2 and 3 for linear and quadraic funcions. EXAMPLE 7 Infecious Diseases The U.S. Deparmen of Healh and Human Services published he daa in Table 1. Table 1 Repored Cases of Infecious Diseases Year Mumps Rubella ,953 56, ,576 3, ,292 1, An exponenial model for he daa on mumps is given by N 91,400(0.835) where N is he number of repored cases of mumps and is ime in years wih 0 represening (A) Use he model o predic he number of repored cases of mumps in (B) Compare he acual number of cases of mumps repored in 1980 o he number given by he model.

12 480 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS SOLUTIONS (A) The year 2010 is represened by 40. Evaluaing N 91,400(0.835) a 40 gives a predicion of 67 cases of mumps in (B) The year 1980 is represened by 10. Evaluaing N 91,400(0.835) a 10 gives 15,060 cases in The acual number of cases repored in 1980 was 8,576, nearly 6,500 less han he number given by he model. Technology Connecions Figure 6 shows he deails of consrucing he exponenial model of Example 7 on a graphing calculaor. 110, (a) Enering he daa Z Figure 6 (b) Finding he model 10,000 (c) Graphing he daa and he model MATCHED PROBLEM 7 An exponenial model for he daa on rubella in Table 1 is given by N 44,500(0.815) where N is he number of repored cases of rubella and is ime in years wih 0 represening (A) Use he model o predic he number of repored cases of rubella in (B) Compare he acual number of cases of rubella repored in 1980 o he number given by he model.

13 S E C T I O N 5 2 Exponenial Models 481 EXAMPLE 8 AIDS Cases and Deahs The U.S. Deparmen of Healh and Human Services published he daa in Table 2. Table 2 Acquired Immunodeficiency Syndrome (AIDS) Cases and Deahs in he Unied Saes Cases Known Diagnosed Deahs Year o Dae o Dae ,185 12, ,755 62, , , , , , , , , , ,060 A logisic model for he daa on AIDS cases is given by 948,000 N e where N is he number of AIDS cases diagnosed by year wih 0 represening (A) Use he model o predic he number of AIDS cases diagnosed by (B) Compare he acual number of AIDS cases diagnosed by 2003 o he number given by he model. SOLUTIONS (A) The year 2010 is represened by 25. Evaluaing 948,000 N e a 25 gives a predicion of approximaely 942,000 cases of AIDS diagnosed by (B) The year 2003 is represened by 18. Evaluaing 948,000 N e a 18 gives 895,013 cases in The acual number of cases diagnosed by 2003 was 929,985, nearly 35,000 greaer han he number given by he model.

14 482 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Technology Connecions Figure 7 shows he deails of consrucing he logisic model of Example 7 on a graphing calculaor. 1,000, (a) Enering he daa (b) Finding he model (c) Graphing he daa and he model Z Figure 7 0 MATCHED PROBLEM 8 A logisic model for he daa on deahs from AIDS in Table 2 is given by 520,000 N e where N is he number of known deahs from AIDS by year wih 0 represening (A) Use he model o predic he number of known deahs from AIDS by (B) Compare he acual number of known deahs from AIDS by 2003 o he number given by he model. Z A Comparison of Exponenial Growh Phenomena The equaions and graphs given in Table 3 compare he growh models discussed in Examples 1 hrough 8. Following each equaion and graph is a shor, incomplee lis of areas in which he models are used. In he firs case (unlimied growh), y S as S. In he oher hree cases (exponenial decay, limied growh, and logisic growh), he graph approaches a horizonal asympoe as S ; hese asympoes (y 0, y c, and y M, respecively) are easily deduced from he given equaions. Table 3 only ouches on a subjec ha you are likely o sudy in greaer deph in he fuure.

15 S E C T I O N 5 2 Exponenial Models 483 Table 3 Exponenial Growh and Decay Descripion Equaion Graph Uses Unlimied growh y ce k c, k 0 y Shor-erm populaion growh (people, baceria. ec.); growh of money a coninuous compound ineres c 0 Exponenial decay y ce k c, k 0 c y Radioacive decay; ligh absorpion in waer, glass, and he like; amospheric pressure; elecric circuis 0 Limied growh y c(1 e k ) c, k 0 c y Learning skills; sales fads; company growh; elecric circuis 0 Logisic growh M y y Long-erm populaion growh; epidemics; sales of 1 ce k new producs; company growh c, k, M 7 0 M 0 ANSWERS TO MATCHED PROBLEMS 1. (A) 1,320 baceria (B) 4,100, baceria 2. (A) 50 baceria (B) 12,000 baceria 3. (A) 43.9 milligrams (B) 8.12 milligrams milligrams 5. (A) 143,000 viewers; 619,000 viewers (B) N approaches an upper limi of 2 million, he number of poenial viewers 6. (A) 48 individuals; 353 individuals (B) N approaches an upper limi of 400, he number of people in he enire group. 7. (A) 12 cases (B) The acual number of cases was 1,850 less han he number given by he model. 8. (A) 519,000 deahs (B) The acual number of known deahs was approximaely 21,000 greaer han he number given by he model.

16 484 CHAPTER 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 5-2 Exercises APPLICATIONS 1. GAMING A person bes on red and black on a roulee wheel using a Maringale sraegy. Tha is, a $2 be is placed on red, and he be is doubled each ime unil a win occurs. The process is hen repeaed. If black occurs n imes in a row, hen L 2 n dollars is los on he nh be. Graph his funcion for 1 n 10. Alhough he funcion is defined only for posiive inegers, poins on his ype of graph are usually joined wih a smooh curve as a visual aid. 2. BACTERIAL GROWTH If baceria in a cerain culure double 1 every 2 hour, wrie an equaion ha gives he number of baceria N in he culure afer hours, assuming he culure has 100 baceria a he sar. Graph he equaion for POPULATION GROWTH Because of is shor life span and frequen breeding, he frui fly Drosophila is used in some geneic sudies. Raymond Pearl of Johns Hopkins Universiy, for example, sudied 300 successive generaions of descendans of a single pair of Drosophila flies. In a laboraory siuaion wih ample food supply and space, he doubling ime for a paricular populaion is 2.4 days. If we sar wih 5 male and 5 female flies, how many flies should we expec o have in (A) 1 week? (B) 2 weeks? 4. POPULATION GROWTH If Kenya has a populaion of abou 34,000,000 people and a doubling ime of 27 years and if he growh coninues a he same rae, find he populaion in (A) 10 years (B) 30 years Compue answers o 2 significan digis. 5. INSECTICIDES The use of he insecicide DDT is no longer allowed in many counries because of is long-erm adverse effecs. If a farmer uses 25 pounds of acive DDT, assuming is half-life is 12 years, how much will sill be acive afer (A) 5 years? (B) 20 years? Compue answers o wo significan digis. 6. RADIOACTIVE TRACERS The radioacive isoope echneium-99m ( 99m Tc) is used in imaging he brain. The isoope has a half-life of 6 hours. If 12 milligrams are used, how much will be presen afer (A) 3 hours? (B) 24 hours? Compue answers o hree significan digis. 7. POPULATION GROWTH If he world populaion is abou 6.5 billion people now and if he populaion grows coninuously a a relaive growh rae of 1.14%, wha will he populaion be in 10 years? Compue he answer o wo significan digis. 8. POPULATION GROWTH If he populaion in Mexico is around 106 million people now and if he populaion grows coninuously a a relaive growh rae of 1.17%, wha will he populaion be in 8 years? Compue he answer o hree significan digis. 9. POPULATION GROWTH In 2005 he populaion of Russia was 143 million and he populaion of Nigeria was 129 million. If he populaions of Russia and Nigeria grow coninuously a relaive growh raes of 0.37% and 2.56%, respecively, in wha year will Nigeria have a greaer populaion han Russia? 10. POPULATION GROWTH In 2005 he populaion of Germany was 82 million and he populaion of Egyp was 78 million. If he populaions of Germany and Egyp grow coninuously a relaive growh raes of 0% and 1.78%, respecively, in wha year will Egyp have a greaer populaion han Germany? 11. SPACE SCIENCE Radioacive isoopes, as well as solar cells, are used o supply power o space vehicles. The isoopes gradually lose power because of radioacive decay. On a paricular space vehicle he nuclear energy source has a power oupu of P was afer days of use as given by P 75e Graph his funcion for EARTH SCIENCE The amospheric pressure P, in pounds per square inch, decreases exponenially wih aliude h, in miles above sea level, as given by P 14.7e 0.21h Graph his funcion for 0 h MARINE BIOLOGY Marine life is dependen upon he microscopic plan life ha exiss in he phoic zone, a zone ha goes o a deph where abou 1% of he surface ligh sill remains. Ligh inensiy I relaive o deph d, in fee, for one of he cleares bodies of waer in he world, he Sargasso Sea in he Wes Indies, can be approximaed by I I 0 e d

17 S E C T I O N 5 2 Exponenial Models 485 where I 0 is he inensiy of ligh a he surface. To he neares percen, wha percenage of he surface ligh will reach a deph of (A) 50 fee? (B) 100 fee? 14. MARINE BIOLOGY Refer o Problem 13. In some waers wih a grea deal of sedimen, he phoic zone may go down only 15 o 20 fee. In some murky harbors, he inensiy of ligh d fee below he surface is given approximaely by I I 0 e 0.23d Wha percenage of he surface ligh will reach a deph of (A) 10 fee? (B) 20 fee? 15. AIDS EPIDEMIC The World Healh Organizaion esimaed ha 39.4 million people worldwide were living wih HIV in Assuming ha number coninues o increase a a relaive growh rae of 3.2% compounded coninuously, esimae he number of people living wih HIV in (A) 2010 (B) AIDS EPIDEMIC The World Healh Organizaion esimaed ha here were 3.1 million deahs worldwide from HIV/AIDS during he year Assuming ha number coninues o increase a a relaive growh rae of 4.3% compounded coninuously, esimae he number of deahs from HIV/AIDS during he year (A) 2008 (B) NEWTON S LAW OF COOLING This law saes ha he rae a which an objec cools is proporional o he difference in emperaure beween he objec and is surrounding medium. The emperaure T of he objec hours laer is given by T T m (T 0 T m )e k where T m is he emperaure of he surrounding medium and T 0 is he emperaure of he objec a 0. Suppose a bole of wine a a room emperaure of 72 F is placed in he refrigeraor o cool before a dinner pary. If he emperaure of in he refrigeraor is kep a 40 F and k 0.4, find he emperaure of he wine, o he neares degree, afer 3 hours. (In Exercise 5-5 we will find ou how o deermine k.) 18. NEWTON S LAW OF COOLING Refer o Problem 17. Wha is he emperaure, o he neares degree, of he wine afer 5 hours in he refrigeraor? 19. PHOTOGRAPHY An elecronic flash uni for a camera is acivaed when a capacior is discharged hrough a filamen of wire. Afer he flash is riggered, and he capacior is discharged, he circui (see he figure) is conneced and he baery pack generaes a curren o recharge he capacior. The ime i akes for he capacior o recharge is called he recycle ime. For a paricular flash uni using a 12-vol baery pack, he charge q, in coulombs, on he capacior seconds afer recharging has sared is given by q (1 e 0.2 ) Find he value ha q approaches as increases wihou bound and inerpre. I R V 20. MEDICINE An elecronic hear pacemaker uses he same ype of circui as he flash uni in Problem 19, bu i is designed so ha he capacior discharges 72 imes a minue. For a paricular pacemaker, he charge on he capacior seconds afer i sars recharging is given by q (1 e 2 ) Find he value ha q approaches as increases wihou bound and inerpre. 21. WILDLIFE MANAGEMENT A herd of 20 whie-ailed deer is inroduced o a coasal island where here had been no deer before. Their populaion is prediced o increase according o he logisic curve 100 N 1 4e 0.14 where N is he number of deer expeced in he herd afer years. (A) How many deer will be presen afer 2 years? Afer 6 years? Round answers o he neares ineger. (B) How many years will i ake for he herd o grow o 50 deer? Round answer o he neares ineger. (C) Does N approach a limiing value as increases wihou bound? Explain. 22. TRAINING A rainee is hired by a compuer manufacuring company o learn o es a paricular model of a personal compuer afer i comes off he assembly line. The learning curve for an average rainee is given by 200 N 4 21e 0.1 (A) How many compuers can an average rainee be expeced o es afer 3 days of raining? Afer 6 days? Round answers o he neares ineger. (B) How many days will i ake unil an average rainee can es 30 compuers per day? Round answer o he neares ineger. (C) Does N approach a limiing value as increases wihou bound? Explain. C S