Exponential Growth and Decay

Transcription

1 Connecing Algebra o Advanced Placemen* Mahemaics A Resource and Sraegy Guide Exponenial Growh and Decay Objecive: To give sudens pracice in solving he ype of exponenial growh and decay problems similar o hose hey will encouner on he AP Calculus exam. Connecions o Previous Learning: Sudens should be familiar wih solving exponenial growh and decay problems using he universal growh formula y. Connecions o AP*: AP Calculus Topics: Analysis of Funcions Maerials: Suden Aciviy pages, graphing calculaors Teacher Noes: Many Algebra exboos limi examples of hese ypes of problems o bacerial growh and radioacive decay using bases of and ½ respecively. They migh also include examples of coninuously compounding ineres. The purpose of his lesson is o broaden he range of applicaions o any siuaion where he rae of growh in somehing is direcly proporional o he amoun of ha somehing presen. Mos of hese problems were adaped direcly from old AP quesions in order o beer familiarize sudens wih he wording and syle of quesioning used on he AP exams. Problem #7 is paricularly ricy for sudens since hey are no given specific values wih which o wor. Teachers may wan sudens o choose differen arbirary values o firs wor he problem and hen ry o deermine how o generalize ha approach. The las wo quesions on he worshee involve ransformaions of he universal growh formula and as sudens o use muliple represenaions in order o predic fuure behavior. *Advanced Placemen and AP are regisered rademars of he College Enrance Examinaion Board. The College Board was no involved in he producion of his produc. Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi: 1

2 Suden Aciviy Exponenial Growh and Decay 1. A any ime 0, in days, he number of baceria presen y is given by y where is a consan. The iniial populaion is 1,000 and he populaion riples during he firs days. a) Wrie an expression for y a any ime 0. b) By wha facor will he populaion have increased in he firs 10 days? c) A wha ime, in days, will he populaion have increased by a facor of 6?. The rae of consumpion of cola in he Unied Saes is given by S( ) where S is measured in billions of gallons per year and is measured in years from he beginning of a) The consumpion rae doubles every years and he consumpion rae a he beginning of 1980 was 6 billion gallons per year. Find C and. b) Using he informaion from a, wha will he consumpion rae of cola be in 1987? c) Using he informaion from a, when would you predic ha he consumpion rae will exceed 10 billion gallons per year? 3. A puppy weighs.0 pounds a birh and 3. pounds wo monhs laer. If he weigh of he puppy during is firs 6 monhs is increasing a a rae proporional o is weigh, hen how much will he puppy weigh when i is 3 monhs old? 4. During a cerain epidemic, he number of people ha are infeced a any ime increases a a rae proporional o he number of people ha are infeced a ha ime. If 1,000 people are infeced when he epidemic is firs discovered, and 100 are infeced 7 days laer, how many people are infeced 1 days afer he epidemic is firs discovered?. Oil is being pumped coninuously from a cerain oil well a a rae proporional o he amoun of oil lef in he well, where y is he amoun of oil lef in he well a any ime. Iniially here were 1,000,000 gallons of oil in he well, and 6 years laer here were 00,000 gallons remaining. I will no longer be profiable o pump oil when here are fewer han 0,000 gallons remaining. a) Wrie an equaion for y, he amoun of oil remaining in he well a any ime. b) A wha rae is he amoun of oil in he well decreasing when here are 600,000 gallons of oil remaining? ( R() = y ) c) In order no o lose money, a wha ime should oil no longer be pumped from he well? Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi:

3 Suden Aciviy 6. Buzz Lighyear is reurning o Earh in his spaceship when he deecs an oxygen lea. He nows ha he rae of change of he pressure is direcly proporional o he pressure of he remaining oxygen. a) Wrie an equaion ha expresses his fac. b) Solve your equaion subjec o he iniial condiion ha he pressure is 3000 psi a ime = 0 when Buzz discovers he lea. c) Five hours afer he discovers he lea, he pressure has dropped o 300 psi. A ha ime, Buzz is sill 1 hours away from Earh. Will he mae i home before he pressure drops o 800 psi? 7. Baceria in a cerain culure increases a a rae proporional o he number presen. If he number of baceria doubles in hree hours, in how many hours will he number of baceria riple? 8. Le P() represen he number of wolves in a populaion a ime years, when 0. The populaion of wolves is given by P = Ce. ( ) 800 a) If P(0) = 00, find P() in erms of and. b) If P() = 700, find. c) As ime increases wihou bound, wha happens o he populaion of wolves? Suppor your answer wih boh a able of values and a graph. 9. Le v() be defined by v( ) ( ) 16 he velociy, in fee per second, of a sydiver a ime seconds. a) Wih he condiion v(0) = -0, find an expression for v in erms of where is measured in seconds. b) I is safe o land when he sydiver's velociy is -0 fee per second. A wha ime does she reach his velociy? c) When he objec reaches erminal velociy, he acceleraion is zero; in oher words, i is no longer changing speeds. Use a able of values and a graph o deermine he erminal velociy for he sydiver. Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi: 3

4 Connecing Algebra o Advanced Placemen* Mahemaics A Resource and Sraegy Guide Exponenial Growh and Decay Answers: 1. y Ce ( y) ( ) ( =, 0,1000 and,3000) = C 3000 = 100e ( ) 3 = e 3 = 3 = y = 1000e y= 1000e y = 1000e y = 1000e 3 9 ( ) y = y = 9000 Afer 10 days, he populaion will have increased by a facor of = 1000e 6 = e = 6 = 3 = 8.1 days Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi: 4

5 . a) S() ( 0, 6) (,1) 6 6 = C 1 = 6e = e ( ) ( ) ( ) = = b) S() = 6e 7 ( ) = In 1987, he consumpion rae of cola would be billion gallons of soda per year. c) 10 = 6e = e = = 3.19 = According o he model, he consumpion rae will exceed 10 billion gallons per year in 003. Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi:

6 3. W ( ) ( 0,.0) and (,3.).0.0 = C 3. =.0e ( ) ( ) 1.7 = e 1.7 = = W = e W = Afer 3 monhs, he puppy will weigh lbs. 4. P Ce ( ) ( ) ( = 0,1000 7,100 1,? ) = C 100 = 1000e ( 7) 1. = e 1. = = P= 1000e ( 7) P = Afer 1 days, abou 1367 people will be infeced.. a) y Ce ( ) ( = 0, ,0000) = C 0000 = e. =. 6 e ( 6) = y = e. 6 ( 6) Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi: 6

7 b) R ( ) = y. R = 6 R () = 69, () ( ) Oil in he well is decreasing a a rae of 69,31 gallons per year when here are 600,000 gallons remaining. c) 0, 000 = e. 6.0 = e = =..93 years = I will no longer be profiable when he well is down o 0,000 gallons of oil which will ae abou 6 years. 6. a) b) c) P = C P = 3000e 300 = 3000e 3 30 = e ( ) 3 = = = 3000e 4 1 = e ( ) = 1 30 Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi: 7

8 4 1 = = Yes, i will ae hours for he pressure o drop o 800psi, and Buzz will be home in only 0 hours. Buzz will mae i home! B 0, C 3, C?,3C) 7. ( ) ( ) ( C ( 3) ( 3) = e = 3 3 3C 3 = e = 3 3 () 3 = 3 33 = 4.7 hours = Baceria will riple afer 4.7 hours. 8. a) 00 = 800 Ce () 00 = 800 C = 1C 300 = C P () = e Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi: 8

9 b) 700 = e 100 = = e 1 = 3 e 1 3 = 3 = 3 = ( ) c) P() = e P( ) , As ime increases, he populaion of wolves sabilizes a around 800 wolves. 9. a) V() 16 ( ) = 34 = C () 1 34 = C 0 0 Ce 16 () e V = Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi: 9

10 b) = 0 34e 16 4= = e e = = = I is safe o land afer abou seconds. () e V = V( ) Terminal velociy for he sydiver is 16 f/sec. Copyrigh 009 Laying he Foundaion, Inc. Dallas, TX. All righs reserved. Visi: 10