Growth and Decay (pp. 1 of 5)

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1 Growh and Deca (pp. of 5) HS Mahemaics Uni: 9 Lesson: The general formula shown here can be used o describe quaniies ha increase ( ) A ( r ) or decrease a a cerain percenage rae each ear. In his model, A = he original amoun; r = he annual percenage rae of increase (+) or decrease (-); = ime (in ears); and () = he amoun afer ears. Following his formula, complee he equaions and ables o answer he quesions ha follow.. Righ now, he populaion of our own is around 5,. However, he number of ciizens is projeced o increase abou 8% ever ear. Wha will he populaion be in ears? Funcion: = ( ) Time (ears) Populaion. A manufacuring plan sends 5, cubic fee of CO ino he amosphere annuall, bu has signed an environmenal agreemen o cu hese emissions b % each ear. To wha level will he carbon emissions drop in ears? Funcion: = ( ) Time (ears) CO Emissions (f ). Mr. McDonald jus bough a farm racor for $6,. For a purposes, is value is epeced o depreciae a a rae of 8% annuall. Wha will he racor s value be in ears? Funcion: = ( ) Time (ears) Value ($). For her college fund, Jenn s parens invesed $5 four ears ago. The accoun has now reached $6. Wha is he approimae annual reurn rae (r) on his invesmen? Funcion: = ( ) Time (rs) Invesmen amoun ($) $5, $5, $5,68 $5,955 $6,

2 Growh and Deca (pp. of 5) HS Mahemaics Uni: 9 Lesson: ( ) A ( r ) ( ) A ( ) r n n Again, his general formula can be used o describe quaniies ha increase a a cerain percenage rae each ear. In finance, his increase is called ineres. However, when banks handle invesmens (or loans) he don simpl appl ineres once a ear. Insead, he give ou a lile ineres hroughou he ear. When a porion of an invesmen s ineres is given a inervals hroughou he ear, his differen formula mus be used, where n = he number of imes ineres is given in a ear. 5. Here, $6, is invesed a a % annual ineres rae for 5 ears. How will he invesmen change if par of he ineres is given ahead of ime? * Annuall (once a ear) n = = 6 ( +. )^ Years Amoun $6, $6, $6,89.6 $6,79.8 $7,9.5 5 $7,99.9 A) Quarerl (four imes a ear) n = = (+ / )^( ) Years Amoun $6, 5 B) Monhl n = = (+ / )^( ) Years Amoun $6, 5 C) Wha would be in he accoun afer 5 ears if a lile of he ineres is given ever week? N =, so = (+ / )^( * 5 ) D) Wha if ineres accrues dail? N =, so = (+ / )^( * 5 )

3 Growh and Deca (pp. of 5) HS Mahemaics Uni: 9 Lesson: Coninue o complee he ables and quesions below. 6. Funcion: = 5(.8) Time Mass (ears) (grams) Radioacive maerial loses is mass over ime because some of is nuclear maerial is released as radiaion. The mass of a cerain sample changes according he equaion o he lef. Afer approimael how man ears does he sample have onl half of is original mass? This is called he maerial s half-life. Has all he subsance gone afer wo half-lives? Eplain. 7. Funcion: = ( ) Time Mass (ears) (grams) A differen radioacive subsance loses half is mass over a 6- ear period of ime (or, is half-life is 6 ears). A -gram sample will deca ino a 6-gram sample afer 6 ears. Deermine he approimae rae a which he subsance decas per ear. Then wrie he equaion for is mass () over ime () in ears. Wih our equaion, find () and (8). Do hese values make sense? For wha inerval of ime will he amoun of he subsance be greaer han grams? Solve graphicall or b able.

4 Growh and Deca (pp. of 5) HS Mahemaics Uni: 9 Lesson: You ma have noiced ha he las quesion (problem 7) acuall deals wih an inequali: For wha inerval of ime will he amoun of he subsance be greaer han grams? Translaes ino () >, or (.89) > You can use he following seps on a graphing calculaor o find he soluions o eponenial inequaliies, such as (.89) Ener he lef side of he inequali ino Y. Y = (.89)^X Ener he righ side of he inequali ino Y. Y = Graph in a good WINDOW o deermine where he inequali holds rue. Check wih a able. WINDOW Xmin= GRAPH According o he graph and able, he inequali Xma= 5 (.89) is rue Xscl= when <9.5. Ymin= Also, since he acual funcion Yma= onl uses, he soluion is Yscl= beer saed as <9.5. TABLE Tr solving hese inequaliies on our own. 8. 5(.) 9. (.5) 5 Y = Y =. 9 Y = Y = Y = Skech he GRAPH. Skech he GRAPH. Skech he GRAPH. Y = Check wih a TABLE. Y Y Check wih a TABLE. Y Y Check wih a TABLE. Y Y Final Answer: Final Answer: Final Answer:

5 Growh and Deca (pp. 5 of 5) HS Mahemaics Uni: 9 Lesson: Coninue o complee he ables and quesions below. One of he remaining iems ma have inequaliies. Read carefull!. Funcion: = 5(.78).5 Time Populaion (ears) (# rabbis) 5 When a bunch of rabbis were se free in a residenial area, he animals had no naural predaors, and so he populaion grew rapidl, according o he equaion a he lef. Compared o is original size, how man imes greaer was he populaion of rabbis afer 5 ears? According o he equaion, wha would be he populaion of he rabbis a = ears? Does he same scale facor appl? For wha inerval of ime will he number of rabbis eceed,?. Funcion: = 55(.7) + 75 Time Temperaure (minues) (F) Mr. Java pours a cup of ho coffee, hen ses i aside and forges abou i. Over he ne several minues, he coffee cools down according o he equaion a he lef. Find he raio beween () and (). Find he raio beween (7) and (6). Ordinaril, hese raios would be equal o b (which is.7 in his problem). However, hese raios are differen. Wh? Find (5) and (). Wh do ou hink hese answers are so close ogeher?

6 Growh and Deca (pp. of 5) KEY HS Mahemaics Uni: 9 Lesson: The general formula shown here can be used o describe quaniies ha increase ( ) A ( r ) or decrease a a cerain percenage rae each ear. In his model, A = he original amoun; r = he annual percenage rae of increase (+) or decrease (-); = ime (in ears); and () = he amoun afer ears. Following his formula, complee he equaions and ables o answer he quesions ha follow.. Righ now, he populaion of our own is around 5,. However, he number of ciizens is projeced o increase abou 8% ever ear. Wha will he populaion be in ears? Around, people Funcion: = _5_ (.8 ) Time (ears) Populaion 5, 7, 9,6,9,. A manufacuring plan sends 5, cubic fee of CO ino he amosphere annuall, bu has signed an environmenal agreemen o cu hese emissions b % each ear. To wha level will he carbon emissions drop in ears? Abou 9,985 cu.f. Funcion: = _5,_ (.88) Time (ears) CO Emissions (f ) 5,, 8,7,7 9,985. Mr. McDonald jus bough a farm racor for $6,. For a purposes, is value is epeced o depreciae a a rae of 8% annuall. Wha will he racor s value be in ears? Abou $7,7 Funcion: = _6,_ (.8 ) Time (ears) Value ($) $6, $9, $, $,8 $7,7. For her college fund, Jenn s parens invesed $5 four ears ago. The accoun has now reached $6. Wha is he approimae annual reurn rae (r) on his invesmen? Answer: 6% Funcion: = _5,_ (.6) Time (rs) Invesmen amoun ($) Y $5, $5, $5,68 $5,955 $6,

7 Growh and Deca (pp. of 5) KEY HS Mahemaics Uni: 9 Lesson: ( ) A ( r ) ( ) A ( ) r n n Again, his general formula can be used o describe quaniies ha increase a a cerain percenage rae each ear. In finance, his increase is called ineres. However, when banks handle invesmens (or loans) he don simpl appl ineres once a ear. Insead, he give ou a lile ineres hroughou he ear. When a porion of an invesmen s ineres is given a inervals hroughou he ear, his differen formula mus be used. Here, n = he number of imes ineres is given in a ear. 5. Here, $6, is invesed a a % annual ineres rae for 5 ears. How will he invesmen change if par of he ineres is given ahead of ime? * Annuall (once a ear) n = = 6 ( +. )^ Years Amoun $6, $6, $6,89.6 $6,79.8 $7,9.5 5 $7,99.9 A) Quarerl (four imes a ear) n = = 6 (+. / _ )^( _) Years Amoun $6, $6,.6 $6,97. $6,76.95 $7,5.7 5 $7,. B) Monhl n = = 6 (+. / _ )^( ) Years Amoun $6, $6,. $6,98.85 $6,76.6 $7,9.9 5 $7,5.97 C) Wha would be in he accoun afer 5 ears if a lile of he ineres is given ever week? N = _5_, so = 6 (+. / 5 )^( 5* 5 ) $77.85 D) Wha if ineres accrues dail? N = _65_, so = 6 (+. / 65 )^( 65* 5 ) $78.

8 Growh and Deca (pp. of 5) KEY HS Mahemaics Uni: 9 Lesson: Coninue o complee he ables and quesions below. 6. Funcion: = 5(.8) Time Mass (ears) (grams) Radioacive maerial loses is mass over ime because some of is nuclear maerial is released as radiaion. The mass of a cerain sample changes according he equaion o he lef. Afer approimael how man ears does he sample have onl half of is original mass? This is called he maerial s half-life. Afer ears, half he original 5 g remains. Has all he subsance gone afer wo half-lives? Eplain. No. Afer wo half-lives (or, 8 ears), 5 g remains. This is half of half he original amoun (or one-fourh as much). 7. Funcion: = (.89) Time Mass (ears) (grams) A differen radioacive subsance loses half is mass over a 6- ear period of ime (or, is half-life is 6 ears). For eample, a -gram sample will deca ino a 6-gram sample afer 6 ears. Deermine he approimae rae a which he subsance decas per ear. Then wrie he equaion for is mass () over ime () in ears. The raio beween erms is abou.89. Equaion: = (.89) Wih our equaion, find () and (8). Do hese values make sense? (), and (8) 5. Yes; half he subsance remains afer ever 6-ear period. For wha inerval of ime will he amoun of he subsance be greaer han grams? Solve graphicall or b able. I will be above grams from a ime of o approimael 9.5 ears. 7

9 Growh and Deca (pp. of 5) KEY HS Mahemaics Uni: 9 Lesson: You ma have noiced ha he las quesion (problem 7) acuall deals wih an inequali: For wha inerval of ime will he amoun of he subsance be greaer han grams? Translaes ino () >, or (.89) > You can use he following seps on a graphing calculaor o find he soluions o eponenial inequaliies, such as (.89) Ener he lef side of he inequali ino Y. Y = (.89)^X Ener he righ side of he inequali ino Y. Y = Graph in a good WINDOW o deermine where he inequali holds rue. Check wih a able. WINDOW Xmin= GRAPH According o he graph and able, he inequali Xma= 5 (.89) is rue Xscl= when <9.5. Ymin= Also, since he acual funcion Yma= onl uses, he soluion is Yscl= beer saed as <9.5. TABLE Tr solving hese inequaliies on our own. 8. 5(.) 9. (.5) 5 Y = 5(.)^ Y = (.5)^. Y = / 9 Y = Y = 5 Y = 9^ Skech he GRAPH. Skech he GRAPH. Skech he GRAPH. Check wih a TABLE. Y Y Final Answer: Check wih a TABLE. Y Y Final Answer: 8. Check wih a TABLE. Y Y Final Answer: -.5

10 Growh and Deca (pp. 5 of 5) KEY HS Mahemaics Uni: 9 Lesson: Coninue o complee he ables and quesions below. One of he remaining iems ma have inequaliies. Read carefull!. Funcion: = 5(.78).5 Time Populaion (ears) (# rabbis) When a bunch of rabbis were se free in a residenial area, he animals had no naural predaors, and so he populaion grew rapidl, according o he equaion a he lef. Compared o is original size, how man imes greaer was he populaion of rabbis afer 5 ears? 69/5=.8, so abou imes greaer According o he equaion, wha would be he populaion of he rabbis a = ears? Does he same scale facor appl? A =, he populaions is 7 rabbis. Yes, his is approimael he same as 69. For wha inerval of ime will he number of rabbis eceed,? >.8 The populaion will eceed, rabbis when he number of ears since he rabbis were released is greaer han.8 ears.. Funcion: = 55(.7) + 75 Time Temperaure (minues) (F) Mr. Java pours a cup of ho coffee, hen ses i aside and forges abou i. Over he ne several minues, he coffee cools down according o he equaion a he lef. Find he raio beween () and ()..5/ =.87 Find he raio beween (7) and (6)..5/ =.975 Ordinaril, hese raios would be equal o b (which is.7 in his problem). However, hese raios are differen. Wh? The plus 75 in he equaion makes he raios no be equivalen. Find (5) and (). Wh do ou hink hese answers are so close ogeher? (5) 75.6F; () 75.F. As ime goes on he emperaure of he coffee nears 75F. 7