High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 1. You have see this fomula fo compoud ieest. Paamete P is the picipal amou (the moey you stat with). Paamete is the ieest ate pe yea expessed as a decimal (5% is 05). Paamete is the umbe of times the ieest is compouded pe yea. Idepede vaiable t is the umbe of yeas the picipal has bee i the accou. The fuctio A is the amou of moey i the accou afte time t (i yeas). Which fomula shows how the compoud ieest fomula is simplified whe ieest is oly compouded oce pe yea ( = 1)? A A(t) = P(1 t ) B A(t) = P(1 ) C A(t) = P(1 ) t D A(t) = P(1 t) 2. I the fomula A(t) = P(1 ) t the gowth facto is 1. What is the decimal value of whe the ieest ate is 3.65%? = 3. Oe model fo calculatig ieest uses this fomula: A(t) = 10,000 (1.00125) t. I this fomula, the picipal is $ ad the ate is %. 4. This fuctio models the gowth of $100 ivested i a accou eaig 3% ieest compouded aually. A(t) = 100(1 03) t = 100(1.03) t Make a table showig the amou i the accou afte 1 y, 51 y, ad 101 y. 5. The fuctio show models the gowth of $100 ivested i a accou eaig 3% ieest compouded quately (4 times pe yea). At () = 100 4t 003. = 100( 1. 0075) 4 4t Make a table showig the amou i the accou afte 1 y, 51 y, ad 101 y. Copyight 2015 Peaso Educatio, Ic. 37
High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 6. Now suppose $100 is ivested i a accou eaig 3% ieest compouded daily. a. Wite a fuctio that models this accou usig the stadad fom. b. Make a table showig the amou i the accou afte 1 y, 51 y, ad 101 y. 7. Coside how compoudig ieest affects moey i the bak. A compay has $400,000 i a bak accou eaig 3% ieest. Recall the fomula fo compoud ieest. a. Wite a fuctio fo the accou balace fo yealy ad quately compoudig = 1 (aually) ad = 4 (quately) ove t yeas. b. Make a table that shows the accou balace betwee = 1 (aually) ad = 4 (quately) fo 1 y, 10 y, ad 25 y. 8. A compay eceives compoud ieest o $400,000 evey moh at a ieest ate of 2.7%. a. Afte 25 y, how much will this accou ea? b. Compae the eaigs fo 2.7% ieest compouded mohly to 3% ieest compouded quately. Which eas moe? 9. Jaso wote this stateme. If the ieest is compouded 4 times as ofte, the the total ieest eaed should be quadupled. Do you agee with Jaso? Explai you easoig usig the example of the $400,000 accou with a 3% ieest ate ad the geeal compoud ieest fomula. Challege Poblem 1 Coiue cosideig how compoudig ieest affects moey i the bak. a. How would you modify the fuctios fo the $400,000 accou with 3% ieest ate if ieest was compouded: Evey day Evey hou Evey miute b. Exploe the accou value at 25 y ad 100 y fo lage ad lage -values. Copyight 2015 Peaso Educatio, Ic. 38
High School: Expoeial Fuctios 1. You have see this fomula fo compoud ieest. I this fomula, A = the amou of moey at time t (i yeas), P = the picipal amou (the moey you stat with), = the ieest ate pe yea expessed as a decimal (5% is 05), ad = the umbe of times ieest is compouded pe yea. You may have oticed that compoudig ieest esults i moe moey. It is ieestig to see what happes if = 365, fo istace. What is the meaig of this value fo? 2. M(t) = (1 0365) t Nt () = Explai the umbes i both fomulas. 0365 3. Hee ae gaphs of two fuctios: M(t) = (1 0365) t ad Nt () = Match each fuctio to its gaph. 40 M(t) = 1.0365 t Nt () = 0365 0365. Moey ($) 30 20 Gaph A Gaph B 10 0 0 25 50 Time (y) 75 100 Copyight 2015 Peaso Educatio, Ic. 39
High School: Expoeial Fuctios 4. The fomula A()= t P becomes much simple if P = 1 ad = 1. What is the meaig of these two assumptios? 1 5. Whe P = 1 ad = 1, you get At ()=. You have see this fomula ealie. 1 You ca simplify this fomula agai by givig the pat a ew ame: B(). 1 Thus, B ( )=. Use a calculato to fid B() fo = 1, 2, 3, 4, ad 5. 6. Explai why the fomula At ()= 1 ca be witte as A(t) = B t. 1 7. If you coiue to evaluate B ( )= appoach a limit. What does it mea that the fuctio appoaches a limit? fo geate values of, the fuctio will 1 8. As appoaches ifiity, B( )= I that case, you get A(t) = e t. a. Daw the gaph of y = B(). b. Use the gaph to fid a estimate fo the value of e. will appoach the famous umbe e. 9. Which statemes ae tue about e? Thee may be moe tha oe tue stateme. A It is a atioal umbe it ca be expessed as the divisio of two ieges. B It is a costa. C It is used to calculate coiuously compouded ieest. D It is commoly used as a vaiable. E Its decimal digits coiue without epeatig it caot be expessed as the divisio of two ieges. Copyight 2015 Peaso Educatio, Ic. 40
High School: Expoeial Fuctios 1 Emma has $800 i a bak accou with a ieest ate of 2.7%. Which equatio models the accou balace afte t yeas? A A(t) = 800 e 1.027t B A(t) = 800 e 973t C A(t) = 800 e 027t D A(t) = 027 e 800t 11. A compay has $100,000 i a bak accou with a ieest ate of 2.7%. Use a calculato to fid how much moey will be i the accou afte 5 y, give coiuously compouded ieest. 12. A compay has $20,000 i a accou eaig 1.35%. How much moey will be i the accou afte 5 y, give coiuously compouded ieest? 13. Kayla was to put moey io a bak accou. She has two optios: she ca ea 4.5% aually o 4% compouded coiuously. Which optio should she choose i ode to maximize he eaigs? Justify you ecommedatio with calculatios ad a gaph. Challege Poblem 14. A fame sows wheat o Apil 1. The wheat will be havested aoud July 3 I the fist 40 days, it gows expoeially. Afte that peiod, the gowth becomes liea. The ate of expoeial gowth ca be descibed by N(t) = 100 e 1(t 40), with 0 t 40, whee t is the umbe of days sice plaig. The ate of liea gowth ca be descibed by N(t) = 100, with 40 < t 10 a. Daw the gaph of N(t) fo 0 t 10 b. At what time is the gowth ate N(t) equal to 10? c. Gaph M(t) = e 1t ad compae the gaph to the gaph of N(t). Explai the diffeeces. Copyight 2015 Peaso Educatio, Ic. 41