Section 4.1 Exercises

Transcription

1 Secion 4.1 Eponenial Funcions 459 Secion 4.1 Eercises For each able below, could he able represen a funcion ha is linear, eponenial, or neiher? f() h() m() g() k() n() A populaion numbers 11,000 organisms iniially and grows by 8.5% each year. Wrie an eponenial model for he populaion. 8. A populaion is currenly 6,000 and has been increasing by 1.2% each day. Wrie an eponenial model for he populaion. 9. The fo populaion in a cerain region has an annual growh rae of 9 percen per year. I is esimaed ha he populaion in he year 2010 was 23,900. Esimae he fo populaion in he year The amoun of area covered by blackberry bushes in a park has been growing by 12% each year. I is esimaed ha he area covered in 2009 was 4,500 square fee. Esimae area ha will be covered in A vehicle purchased for $32,500 depreciaes a a consan rae of 5% each year. Deermine he approimae value of he vehicle 12 years afer purchase. 12. A business purchases $125,000 of office furniure which depreciaes a a consan rae of 12% each year. Find he residual value of he furniure 6 years afer purchase.

2 460 Chaper 4 Find an equaion for an eponenial passing hrough he wo poins 0, 3, (2, 75) 13. 0, 6, (3, 750) , 2000, (2, 20) 16. 0, 9000, (3, 72) ,, 3, ,, 1, ,6, 3, , 4, (3, 2) 21. 3,1, (5, 4) 22. 2,5, (6, 9) 23. A radioacive subsance decays eponenially. A scienis begins wih 100 milligrams of a radioacive subsance. Afer 35 hours, 50 mg of he subsance remains. How many milligrams will remain afer 54 hours? 24. A radioacive subsance decays eponenially. A scienis begins wih 110 milligrams of a radioacive subsance. Afer 31 hours, 55 mg of he subsance remains. How many milligrams will remain afer 42 hours? 25. A house was valued a $110,000 in he year The value appreciaed o $145,000 by he year Wha was he annual growh rae beween 1985 and 2005? Assume ha he house value coninues o grow by he same percenage. Wha will he value equal in he year 2010? 26. An invesmen was valued a $11,000 in he year The value appreciaed o $14,000 by he year Wha was he annual growh rae beween 1995 and 2008? Assume ha he value coninues o grow by he same percenage. Wha will he value equal in he year 2012? 27. A car was valued a $38,000 in he year The value depreciaed o $11,000 by he year Assume ha he car value coninues o drop by he same percenage. Wha will he value be in he year 2013? 28. A car was valued a $24,000 in he year The value depreciaed o $20,000 by he year Assume ha he car value coninues o drop by he same percenage. Wha will he value be in he year 2014? 29. If 4000 dollars is invesed in a bank accoun a an ineres rae of 7 per cen per year, find he amoun in he bank afer 9 years if ineres is compounded annually, quarerly, monhly, and coninuously.

3 Secion 4.1 Eponenial Funcions If 6000 dollars is invesed in a bank accoun a an ineres rae of 9 per cen per year, find he amoun in he bank afer 5 years if ineres is compounded annually, quarerly, monhly, and coninuously. 31. Find he annual percenage yield (APY) for a savings accoun wih annual percenage rae of 3% compounded quarerly. 32. Find he annual percenage yield (APY) for a savings accoun wih annual percenage rae of 5% compounded monhly. 33. A populaion of baceria is growing according o he equaion P () 1600e measured in years. Esimae when he populaion will eceed A populaion of baceria is growing according o he equaion P () 1200e measured in years. Esimae when he populaion will eceed , wih, wih 35. In 1968, he U.S. minimum wage was $1.60 per hour. In 1976, he minimum wage was $2.30 per hour. Assume he minimum wage grows according o an eponenial model w (), where represens he ime in years afer [UW] a. Find a formula for w ( ). b. Wha does he model predic for he minimum wage in 1960? c. If he minimum wage was $5.15 in 1996, is his above, below or equal o wha he model predics. 36. In 1989, research scieniss published a model for predicing he cumulaive number 1980 of AIDS cases (in housands) repored in he Unied Saes: a , where is he year. This paper was considered a relief, since here was a fear he correc model would be of eponenial ype. Pick wo daa poins prediced by he research model a ( ) o consruc a new eponenial model b ( ) for he number of cumulaive AIDS cases. Discuss how he wo models differ and eplain he use of he word relief. [UW] 3

4 462 Chaper You have a chess board as picured, wih squares numbered 1 hrough 64. You also have a huge change jar wih an unlimied number of dimes. On he firs square you place one dime. On he second square you sack 2 dimes. Then you coninue, always doubling he number from he previous square. [UW] a. How many dimes will you have sacked on he 10h square? b. How many dimes will you have sacked on he nh square? c. How many dimes will you have sacked on he 64h square? d. Assuming a dime is 1 mm hick, how high will his las pile be? e. The disance from he earh o he sun is approimaely 150 million km. Relae he heigh of he las pile of dimes o his disance.

5 Secion 4.2 Graphs of Eponenial Funcions 463 Secion 4.2 Eercises Mach each equaion wih one of he graphs below 1. f f f f f f A B C D E F If all he graphs o he righ have equaions wih form f ab 7. Which graph has he larges value for b? A B C D E 8. Which graph has he smalles value for b? F 9. Which graph has he larges value for a? 10. Which graph has he smalles value for a? Skech a graph of each of he following ransformaions of f f g h f f k 2 Saring wih he graph of f 4, wrie he equaion of he graph ha resuls from 17. Shifing f ( ) 4 unis upwards 18. Shifing f ( ) 3 unis downwards 19. Shifing f ( ) 2 unis lef 20. Shifing f ( ) 5 unis righ 21. Reflecing f ( ) abou he -ais 22. Reflecing f ( ) abou he y-ais

6 464 Chaper 4 Describe he long run behavior, as and of each funcion 54 1 f f f f f 28. f 23 1 Find an equaion for each graph as a ransformaion of f Find an equaion for he eponenial graphed

7 Secion 4.3 Logarihmic Funcions 465 Secion 4.3 Eercises Rewrie each equaion in eponenial form 1. log 4( q) m 2. log 3( ) k 3. log ( b a ) c 4. log ( z p ) u 5. log v 6. log r s 7. ln w n 8. ln y Rewrie each equaion in logarihmic form y y a b p v 15. d c k e k 12. h 16. z n y e L Solve for. 17. log log 4( ) log 2( ) log 5( ) log log ln ln 2 Simplify each epression using logarihm properies log log log log log log6 36 log log 10, log log ln e ln e Evaluae using your calculaor 37. log log ln ln 0.02 Solve each equaion for he variable e e e e

8 466 Chaper 4 k Conver he equaion ino coninuous growh f ae 57. f f form f f Conver he equaion ino annual growh f ab f 150e form f 100e f 50e f 80e 65. The populaion of Kenya was 39.8 million in 2009 and has been growing by abou 2.6% each year. If his rend coninues, when will he populaion eceed 45 million? 66. The populaion of Algeria was 34.9 million in 2009 and has been growing by abou 1.5% each year. If his rend coninues, when will he populaion eceed 45 million? 67. The populaion of Seale grew from 563,374 in 2000 o 608,660 in If he populaion coninues o grow eponenially a he same rae, when will he populaion eceed 1 million people? 68. The median household income (adjused for inflaion) in Seale grew from $42,948 in 1990 o $45,736 in If i coninues o grow eponenially a he same rae, when will median income eceed $50,000? 69. A scienis begins wih 100 mg of a radioacive subsance. Afer 4 hours, i has decayed o 80 mg. How long will i ake o decay o 15 mg? 70. A scienis begins wih 100 mg of a radioacive subsance. Afer 6 days, i has decayed o 60 mg. How long will i ake o decay o 10 mg? 71. If $1000 is invesed in an accoun earning 3% compounded monhly, how long will i ake he accoun o grow in value o $1500? 72. If $1000 is invesed in an accoun earning 2% compounded quarerly, how long will i ake he accoun o grow in value o $1300?

9 Secion 4.4 Eercises Simplify using logarihm properies o a single logarihm log 28 log 7 log 32 log log log log log log 3 log (7) log log log 2 log ln 4 ln ln 6 ln log 12 log log 2log 13. 2log 3log log log y 3log z 16. 2log log y log z 1 3 Use logarihm properies o epand each epression log y log ab 5 z c 2 a 19. ln 4 5 b c ln a b 5 c log y log y 23. ln y y 1 y 24. ln log y 3 y log y 3 y 9 2

10 468 Chaper 4 Solve each equaion for he variable e 10e 32. 3e e log log 3(2 4) ln ln log log log log log 4 log log 4log log 5 log log log ( 1) log ( ) log ( 2) log 12log log log 15 log log ln ln 3 ln ln ln 6 ln 6

11 Secion 4.5 Graphs of Logarihmic Funcions 469 Secion 4.5 Eercises For each funcion, find he domain and he verical asympoe f log 5 f log f ln 3 4. f ln 5 5. f log f log f 3log 2 8. f 2log 1 Skech a graph of each pair of funcion f log, g ln f log ( ), g log Skech each ransformaion f 2log f 3ln f ln 14. f log 15. f log 2( 2) 16. f log3 4 Wrie an equaion for he ransformed logarihm graph shown

12 470 Chaper 4 Wrie an equaion for he ransformed logarihm graph shown

13 Secion 4.6 Eponenial and Logarihmic Models 471 Secion 4.6 Eercises 1. You go o he docor and he gives you 13 milligrams of radioacive dye. Afer 12 minues, 4.75 milligrams of dye remain in your sysem. To leave he docor's office, you mus pass hrough a radiaion deecor wihou sounding he alarm. If he deecor will sound he alarm if more han 2 milligrams of he dye are in your sysem, how long will your visi o he docor ake, assuming you were given he dye as soon as you arrived? 2. You ake 200 milligrams of a headache medicine, and afer 4 hours, 120 milligrams remain in your sysem. If he effecs of he medicine wear off when less han 80 milligrams remain, when will you need o ake a second dose? 3. The half-life of Radium-226 is 1590 years. If a sample conains 200 mg, how many milligrams will remain afer 1000 years? 4. The half-life of Fermium-253 is 3 days. If a sample conains 100 mg, how many milligrams will remain afer 1 week? 5. The half-life of Erbium-165 is 10.4 hours. Afer 24 hours a sample has been reduced o a mass of 2 mg. Wha was he iniial mass of he sample, and how much will remain afer 3 days? 6. The half-life of Nobelium-259 is 58 minues. Afer 3 hours a sample has been reduced o a mass of 10 mg. Wha was he iniial mass of he sample, and how much will remain afer 8 hours? 7. A scienis begins wih 250 grams of a radioacive subsance. Afer 225 minues, he sample has decayed o 32 grams. Find he half-life of his subsance. 8. A scienis begins wih 20 grams of a radioacive subsance. Afer 7 days, he sample has decayed o 17 grams. Find he half-life of his subsance. 9. A wooden arifac from an archeological dig conains 60 percen of he carbon-14 ha is presen in living rees. How long ago was he arifac made? (he half-life of carbon-14 is 5730 years) 10. A wooden arifac from an archeological dig conains 15 percen of he carbon-14 ha is presen in living rees. How long ago was he arifac made? (he half-life of carbon-14 is 5730 years)

14 472 Chaper A baceria culure iniially conains 1500 baceria and doubles every half hour. Find he size of he populaion afer: a) 2 hours, b) 100 minues 12. A baceria culure iniially conains 2000 baceria and doubles every half hour. Find he size of he populaion afer: a) 3 hours, b) 80 minues 13. The coun of baceria in a culure was 800 afer 10 minues and 1800 afer 40 minues. a. Wha was he iniial size of he culure? b. Find he doubling period. c. Find he populaion afer 105 minues. d. When will he populaion reach 11000? 14. The coun of baceria in a culure was 600 afer 20 minues and 2000 afer 35 minues. a. Wha was he iniial size of he culure? b. Find he doubling period. c. Find he populaion afer 170 minues. d. When will he populaion reach 12000? 15. Find he ime required for an invesmen o double in value if invesed in an accoun paying 3% compounded quarerly. 16. Find he ime required for an invesmen o double in value if invesed in an accoun paying 4% compounded monhly The number of crysals ha have formed afer hours is given by n 20e How long does i ake he number of crysals o double?. 18. The number of building permis in Pasco years afer 1992 roughly followed he equaion n 400e. Wha is he doubling ime? 19. A urkey is pulled from he oven when he inernal emperaure is 165 Fahrenhei, and is allowed o cool in a 75 room. If he emperaure of he urkey is 145 afer half an hour, a. Wha will he emperaure be afer 50 minues? b. How long will i ake he urkey o cool o 110?

15 Secion 4.6 Eponenial and Logarihmic Models A cup of coffee is poured a 190 Fahrenhei, and is allowed o cool in a 70 room. If he emperaure of he coffee is 170 afer half an hour, a. Wha will he emperaure be afer 70 minues? b. How long will i ake he coffee o cool o 120? 21. The populaion of fish in a farm-socked lake afer years could be modeled by he 1000 equaion P e a. Skech a graph of his equaion b. Wha is he iniial populaion of fish? c. Wha will he populaion be afer 2 years? d. How long will i ake for he populaion o reach 900? 22. The number of people in a own ha have heard rumor afer days can be modeled by 500 he equaion N e. a. Skech a graph of his equaion b. How many people sared he rumor? c. How many people have heard he rumor afer 3 days?? d. How long will i ake 300 people o have heard he rumor? Find he value of he number shown on each logarihmic scale Plo each se of approimae values on a logarihmic scale Inensiy of sounds: Whisper: 10 W / m, Vacuum: 10 W / m, Je: 10 W / m 28. Mass: Amoeba: 5 10 g, Human: 5 10 g, Saue of Libery: 8 10 g The 1906 San Francisco earhquake had a magniude of 7.9 on he MMS scale. A he same ime here was an earhquake wih magniude 4.7 ha caused only minor damage. How many imes more inense was he San Francisco earhquake han he second one?

16 474 Chaper The 1906 San Francisco earhquake had a magniude of 7.9 on he MMS scale. A he same ime here was an earhquake wih magniude 6.5 ha caused less damage. How many imes more inense was he San Francisco earhquake han he second one? 31. One earhquake has magniude 3.9. If a second earhquake has 750 imes as much energy as he firs, find he magniude of he second quake. 32. One earhquake has magniude 4.8. If a second earhquake has 1200 imes as much energy as he firs, find he magniude of he second quake. 33. A colony of yeas cells is esimaed o conain 10 6 cells a ime = 0. Afer collecing eperimenal daa in he lab, you decide ha he oal populaion of cells a ime hours is given by he funcion f 10 e [UW] a. How many cells are presen afer one hour? b. How long does i ake of he populaion o double?. c. Cherie, anoher member of your lab, looks a your noebook and says:...ha formula is wrong, my calculaions predic he formula for he number of yeas 6 cells is given by he funcion. f Should you be worried by Cherie s remark? d. Anja, a hird member of your lab working wih he same yeas cells, ook 6 6 hese wo measuremens: cells afer 4 hours; cells afer 6 hours. Should you be worried by Anja s resuls? If Anja s measuremens are correc, does your model over esimae or under esimae he number of yeas cells a ime? 34. As ligh from he surface peneraes waer, is inensiy is diminished. In he clear waers of he Caribbean, he inensiy is decreased by 15 percen for every 3 meers of deph. Thus, he inensiy will have he form of a general eponenial funcion. [UW] a. If he inensiy of ligh a he waer s surface is I 0, find a formula for I ( d ), he inensiy of ligh a a deph of d meers. Your formula should depend on I0 and d. b. A wha deph will he ligh inensiy be decreased o 1% of is surface inensiy?

17 Secion 4.6 Eponenial and Logarihmic Models Myoglobin and hemoglobin are oygen carrying molecules in he human body. Hemoglobin is found inside red blood cells, which flow from he lungs o he muscles hrough he bloodsream. Myoglobin is found in muscle cells. The funcion p Y M p calculaes he fracion of myoglobin sauraed wih oygen a a 1 p given pressure p orrs. For eample, a a pressure of 1 orr, M(1) = 0.5, which means half of he myoglobin (i.e. 50%) is oygen sauraed. (Noe: More precisely, you need o use somehing called he parial pressure, bu he disincion is no imporan for 2.8 p calculaes he fracion p of hemoglobin sauraed wih oygen a a given pressure p. [UW] a. The graphs of M ( p ) and H( p ) are his problem.) Likewise, he funcion Y Hp given here on he domain 0 p 100; which is which? b. If he pressure in he lungs is 100 orrs, wha is he level of oygen sauraion of he hemoglobin in he lungs? c. The pressure in an acive muscle is 20 orrs. Wha is he level of oygen sauraion of myoglobin in an acive muscle? Wha is he level of hemoglobin in an acive muscle? d. Define he efficiency of oygen ranspor a a given pressure p o be M ( p) H( p). Wha is he oygen ranspor efficiency a 20 orrs? A 40 orrs? A 60 orrs? Skech he graph of M ( p) H( p) ; are here condiions under which ranspor efficiency is maimized (eplain)? 36. The lengh of some fish are modeled by a von Beralanffy growh funcion. For 0.18 L e where L ( ) is Pacific halibu, his funcion has he form he lengh (in cenimeers) of a fish years old. [UW] a. Wha is he lengh of a new-born halibu a birh? b. Use he formula o esimae he lengh of a 6 year old halibu. c. A wha age would you epec he halibu o be 120 cm long? d. Wha is he pracical (physical) significance of he number 200 in he formula for L ()?

18 476 Chaper A cancerous cell lacks normal biological growh regulaion and can divide coninuously. Suppose a single mouse skin cell is cancerous and is mioic cell cycle (he ime for he cell o divide once) is 20 hours. The number of cells a ime grows according o an eponenial model. [UW] a. Find a formula C () for he number of cancerous skin cells afer hours. b. Assume a ypical mouse skin cell is spherical of radius cm. Find he combined volume of all cancerous skin cells afer hours. When will he volume of cancerous cells be 1 cm 3? 38. A ship embarked on a long voyage. A he sar of he voyage, here were 500 ans in he cargo hold of he ship. One week ino he voyage, here were 800 ans. Suppose he populaion of ans is an eponenial funcion of ime. [UW] a. How long did i ake he populaion o double? b. How long did i ake he populaion o riple? c. When were here be 10,000 ans on board? d. There also was an eponenially-growing populaion of aneaers on board. A he sar of he voyage here were 17 aneaers, and he populaion of aneaers doubled every 2.8 weeks. How long ino he voyage were here 200 ans per aneaer? 39. The populaions of ermies and spiders in a cerain house are growing eponenially. The house conains 100 ermies he day you move in. Afer 4 days, he house conains 200 ermies. Three days afer moving in, here are wo imes as many ermies as spiders. Eigh days afer moving in, here were four imes as many ermies as spiders. How long (in days) does i ake he populaion of spiders o riple? [UW]

19 Secion 4.7 Eercises Graph each funcion on a semi-log scale, he find a formula for he linearized funcion in he form log f m b 1. f f f f The graph below is on a semi-log scale, as indicaed. Find an equaion for he eponenial funcion y. ( ) Use regression o find an eponenial equaion ha bes fis he daa given y y y

20 478 Chaper y Toal ependiures (in billions of dollars) in he US for nursing home care are shown below. Use regression o find an eponenial equaion ha models he daa. Wha does he model predic ependiures will be in 2015? Year Ependiure Ligh inensiy as i passes hrough decreases eponenially wih deph. The daa below shows he ligh inensiy (in lumens) a various dephs. Use regression o find an equaion ha models he daa. Wha does he model predic he inensiy will be a 25 fee? Deph (f) Lumen The average price of elecriciy (in cens per kilowa hour) from is given below. Deermine if a linear or eponenial model beer fis he daa, and use he beer model o predic he price of elecriciy in Year Cos The average cos of a loaf of whie bread from is given below. Deermine if a linear or eponenial model beer fis he daa, and use he beer model o predic he price of a loaf of bread in Year Cos