Ma 093 and MA 117A - Exponential Models. Topic 1 Compound Interest

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1 Ma 093 and MA 117A - Eponenial Models Topic 1 Compound Ineres 15) Compound Ineres A person invess $7000 a 10% ineres compounded annuall. a) Find an equaion for he value of he invesmen afer ears. = a* b A= P(1 + r) A= 7000(1 +.10) A= 7000(1.1) = ears afer invesmen was made A = amoun of mone in accoun Coordinaes of poins ( ears, A dollars) WHAT OTHER QUESTIONS CAN WE ASK ABOUT THIS? READ BELOW b) Show graph and label. c) Wha is he -inercep? Wha does i mean wihin cone? If = 0, hen A = The -inercep is The poin is (0 ears, $7000) Meaning: The iniial deposi ino he accoun was $7000 d) Wha is he meaning of b in he equaion = a* b? In his problem, b = 1.1 Inerpre as a rae: The amoun of mone in he accoun increases a a rae of 10% per ear. 1.1 = = 1 + r. Since r = 0.1, hen he rae is 10% per ear. Inerpre as a muliplier: The amoun in he accoun in an ear is obained b mulipling he amoun of he previous ear b 1.1 e) How much will be in he accoun in 8 ears? 8 A = 7000(1.1) = $15, In 8 ears he accoun will have $15, f) When will he amoun in he accoun double? = 7000(1.1) Solve wih he calculaor GRAPHICAL APPROACH Ener Y1 = 7000(1.1) Y2 = Press WINDOW and use [0, 15] for and [0, 15000] for Press GRAPH I will ake abou 7.3 ears o double This is for eam 3 in MATH 093 And for eam 2 in Mah 117A Now solve using algebra: (isolae eponenial and ake logarihms) 2 = 1.1 log 2 = log( 1.1 ) log 2 = *log1.1 log 2 log = =

2 16) Compound Ineres on he da ou were born, our grandparens se a college fund for ou. The deposied $10,000 in an accoun ha paid 8% compounded annuall. How much will ou have available for college when ou urn 18? P = 10,000 r = 0.08 A= P(1 + r) = 18 A = 10000(1 +.08) = 10000(1.08) 18 A = 10000(1.08) = $39, ) The populaion of a counr was 2.5 million people in he ear 2000 and since hen i has been increasing a a rae of 2% annuall. Tha is, each ear, he populaion is abou imes he previous ear s populaion. Wrie an equaion for he populaion of he counr (in millions) a ears since he ear A= P(1 + r) = 2.5 (1.02)^ 18) The revenue from music downloads was $1.98 billion in 2007 and has grown b abou 86% per ear since hen. Tha is, each ear he revenue is abou imes he previous ear s revenue. a) Find an equaion for he revenue, R() (in billions of dollars) in he ear ha is ears since b) Find R(7) and inerpre in cone wih correc unis. (a) R() = 1.98 (1.86)^ (b) R(7) = 1.98(1.86)^7 = This represens he poin (7 ears since 2007, billion dollars) In 2014, he revenue from music downloads reached billion dollars

3 MORE PROBLEMS TO PRACTICE WHICH ARE NOT IN THE HANDOUT 19) Compound Ineres A person invess $10,000 AT 5% ineres compounded annuall. Find an equaion for he value of he invesmen afer ears. = a* b A= P(1 + r) A= 10000(1 +.05) A= 10000(1.05) = ears afer invesmen was made A = amoun of mone in accoun Coordinaes of poins ( ears, A dollars) How much will here be in he accoun afer 15 ears? A = 10000(1.05)^15=$ WHAT ELSE CAN I ASK ABOUT THIS PROBLEM? (1) Graph an eponenial increasing funcion, label aes and he -inercep. (2) Wha is he -inercep? Wha does i mean wihin cone? The -inercep is 10,000. Meaning: Iniiall, we deposied $10000 ino he accoun (3) Wha is he meaning of b in he equaion = a* b? In his problem, b = 1.05 The amoun of mone in he accoun increases a a rae of 5% per ear = = 1 + r. Since r = 0.05, hen he rae is 5% per ear (4) Think on quesions of differen pes: a) TYPE: Given, find A: How much will be in he accoun in 10 ears? 10 A = 10000(1.05) = $16, In 10 ears he accoun will have $16, b) TYPE: Given A, find : When will he amoun have $13, = 10000(1.05)? Solve wih he calculaor GRAPHICAL APPROACH Ener Y1 = 10000(1.05) Y2 = Press WINDOW and use [0, 15] for and [0, 15000] for Press GRAPH In abou 5.4 ears he accoun will have $13,000 Topic 2 Consrucing an eponenial funcion

4 20) The populaion of a counr is growing eponeniall. In he ear 2000 here were 3.5 million people and he populaion doubles ever five ears. Produce an equaion o model he populaion of his counr in erms of ears since = a* b Years since 2000 Populaion (millions of people) (i doubles from 2.5) = 3.5* b 5 7 = 3.5* b divide b = b 1/5 2 = b b = = 3.5*1.149 Pu his in he Y= of he calculaor and look a he TABLE o see if i makes sense, compare he -values for = 0 and = 5 wih he ones on he able above ` The calculaor able shows he poin (5, ) while he able on he op shows (5, 7). Wh is he value differen? Because in he funcion he value of b = has been rounded, i s no eac.

5 21) The number of baceria in is growing eponeniall. Assume ha now here are 2000 baceria and he number riple ever hour. Wrie an equaion o model he number of baceria in he dish in erms of he number of hours from now. X hours from now Y = number of baceria (i riples) = a* b = 2000* b Since he baceria riples EVERY hour, hen, he muliplier is 3 The equaion is: = 2000*3 22) The number of baceria in is growing eponeniall. Assume ha now here are 2000 baceria and he number riple ever 10 hours. Wrie an equaion o model he number of baceria in he dish in erms of he number of hours from now. X hours from now Y = number of baceria (riples) = = a* b 2000* b = 2000*b 3 = b 10 1/10 3 b = b = = 2000*1.116

6 Mah 117A and Mah 093 Inerpreing he coefficien a and he base b of an eponenial funcion = a* b For each of he following problems a) Inerpre in cone he meaning of he coefficien b) Inerpre in cone he meaning of he base: as a muliplier and as a rae. c) Use our knowledge of eponenial funcions and skech he graph. Label he -inercep and he aes in cone. 23) Eample: The funcion P ( ) = 200(1.12) represens he populaion of a counr P, in millions of people a ears since We wan o inerpre in cone he meaning of he coefficien a = 200 and he base b = 1.12 ha are given in he funcion. Recall ha: The coefficien a = 200 represens he -inercep. The poin (0, 200) = (0 ears from 1995, 200 million people) The base b represens he muliplier per ear and also, if ou compare he wo formas of he funcion: = a* b and A= P(1 + r), hen b = 1 + r. since b = 1.12 = , hen, he rae is 0.12 or 12% The inerpreaion follows: In he ear 1995 here were 200 million people in he counr. Ever ear afer ha, he populaion has been increasing b mulipling b 1.12 which implies a rae of 12%per ear. 24) The number of web pages conained in Google s inde, in millions, a ears since 2012 can be modeled wih he funcion G ( ) = 30(1.95). 30 million web pages were conained in Google s inde in he ear Since hen, he number has been increasing b mulipling b 1.95 ever ear which represens a 95% annual rae of increase. 25) The average number of cable channels per household N, a ears since 2010 can be modeled wih he funcion ( ) 135(1.08) N = In 2010, he average number of cable channels per households was 135. Ever ear since hen, ha average has been increasing b mulipling b 1.08 which implies an 8% annual rae.

7 26) The number of TiVo subscribers who go TiVo hrough DIRECTV, in millions, a ears since 2007 can be modeled wih he eponenial funcion ( ) 25(1.50) S = 27) Le N() represen he populaion of a counr in millions a ears since N ( ) = 2.5(1.25) 28) The price of a car, in housands, a ears since bough is given b he model P ( ) = 25(0.87)

8 Eponenial Deca Half Life 29) A sorage ank conains a radioacive elemen. Le p = f() be he amoun (in grams) of he elemen ha remains a ears afer oda. The graph for f is shown below: a) How man grams of he radioacive subsance does he ank conain oda? Use proper unis. The inercep is 200. Toda here are 200 grams of radioacive subsance in he ank. b) Use he graph o deermine he half life of he elemen? Use proper unis. I akes 40 ears for he 200 grams o deca o half (o 100 grams). The half life is 40 ears c) Use he graph o read he coordinaes of 4 poins relaed o he half-life informaion saring wih he Y-inercep. Record he coordinaes on he able. X Y d) Use he graph o esimae he amoun remaining 70 ears from oda. Use proper unis. Esimae he value when is 70. In 70 ears here will be abou 60 grams in he ank e) Use he graph o esimae when he amoun remaining will be 20 grams? Use proper unis. Esimae he value when = 20. I will ake abou 130 ears for he ank o conain 20 grams f) Use he eponenial regression feaure of he calculaor o find he eponenial funcion = a b ha fis he daa. Use he poins from par (c) o ener in he Edior of he calculaor. Round he consans o 3 decimal places.

9 = 200*.983 Le s coninue wih his!!!! WHAT FOLLOWS IS VERY IMPORTANT!!!! INTERPRET a and b WITHIN CONTEXT = 200*.983 Toda, here are 200 grams of radioacive subsance in he ank and ever ear he amoun remaining is muliplied b Noice ha is he same as 98.3% Ever ear, 98.3% of he radioacive subsance remains and 1.7% decas (100% % = 1.7%) Le s use he funcion o check our esimaes o (d) and (e) on he previous page: d) Esimae he amoun remaining 70 ears from oda. Use proper unis. = = * grams a) Esimae when he amoun remaining will be 20 grams? Use proper unis. 20 = 200*.983 Pu in our calculaor Y1 = 200 * 0.983^ Y2 = 20 And use he INTERSECT feaure of he calculaor o find he inersecion I akes abou ears o deca o 20 grams

10 30) A sorage ank conains a radioacive elemen. Le p = f() be he amoun (in grams) of he elemen ha remains a ears afer oda. The graph for f is shown below. (1) Use -scale = -scale = 10 and label he ic-marks along he aes. Each ic mark has a value of 10 (2) Use -scale = 20, -scale = 30 and label he ic-marks along he aes. Use he scales o label he ic-marks on he graph. a) How man grams of he radioacive subsance does he ank conain oda? Use proper unis. The -inercep will be 100 Toda, he ank has 100 grams of radioacive subsance. The -inercep will be 300. There are 300 grams of radioacive subsance in he ank. b) Use he graph o deermine he half life of he elemen? Use proper unis. When = 10, = 50 (50 is ½ of 100) The half-life is 10 ears When = 20, = 150 (150 is ½ of 300) The half-life is 20 ears c) Use he graph o read he coordinaes of 4 poins relaed o he half-life informaion saring wih he Y-inercep. Record he coordinaes on he able d) Use he graph o esimae he amoun remaining 25 ears from oda. Use proper unis. If = 25, ~15 In 25 ears here will be abou 15 grams remaining. If = 25, ~120 In 25 ears here will be abou 120 grams remaining e) Use he graph o esimae when he amoun remaining will be 40 grams? Use proper unis If = 40, ~ 12 In 12 ears here will be abou 40 grams lef If = 40, ~ 60 In 60 ears here will be abou 40 grams lef

11 Problem number 7 coninued: f) Use algebra and he firs wo poins from he able in par (c) o find he eponenial funcion ha fis he daa; round o hree decimal places. X ears from Y = amoun remaining in now grams X ears from Y = amoun remaining in now grams = a* b = 100* b = a* b = 300* b = 100*b = 300*b 0.5 = b = b (1/10) (0.5) = b b = = 100* = b (0.5) 20 = b (1/20) b = = b = 300*0.966 g) Use he poins from par (c) and he eponenial regression feaure of he calculaor o find he eponenial funcion = a b ha fis he daa; round o 3 decimal places. Ener daa, and do EpReg L1, L2, Y1 You should ge he same equaion from par (f) Ener daa, and do EpReg L1, L2, Y1 You should ge he same equaion from par (f) h) Inerpre he consans a and b from he eponenial model following he forma shown on par (g) of problem (6) Toda, here are 300 grams of radioacive Toda, here are 100 grams of radioacive subsance in he ank. Ever ear he amoun subsance in he ank. Ever ear he amoun decas b mulipling b decas b mulipling b Noice ha = 96.6% Noice ha = 93.3% Ever ear, 96.6% of he radioacive subsance Ever ear, 93.3% of he radioacive subsance remains and 3.4% decas. remains and 6.7% decas. (100% % = 3.4%) (100% % = 6.7%)

12 Problem 7 coninued i) Use he model equaion o find he amoun remaining 25 ears from oda. Use proper unis. If = 25, wha is? If = 25, wha is? You do his one You do his one d) Use he model equaion and he graphical approach o deermine when here will be 40 grams in he ank. Graphical Approach i. Wrie he equaion ii. Ener he lef hand side of he equaion in Y1 of he calculaor iii. Ener he righ hand side of he equaion in he Y2 of he calculaor iv. Ener appropriae window values. v. Press 2 nd TRACE [CALC] vi. Selec 5:inersec vii. Press ENTER hree imes unil ou find he poin of inersecion viii. Answer he problem Graphical Approach i. Wrie he equaion ii. Ener he lef hand side of he equaion in Y1 of he calculaor iii. Ener he righ hand side of he equaion in he Y2 of he calculaor iv. Ener appropriae window values. v. Press 2 nd TRACE [CALC] vi. Selec 5:inersec vii. Press ENTER hree imes unil ou find he poin of inersecion viii. Answer he problem You do his one You do his one 40 = 100 * (.933)^ Ener LHS in Y1 Ener RHS in Y2 Use he window {0, 20] for and [0, 100] for Find poin of inersecion 40 = 300 (.966)^ Ener LHS in Y1 Ener RHS in Y2 Use he window {0, 60] for and [0, 300] for Find poin of inersecion Answer: 13.2 ears Answer: 58.2 ears