9.4 Exponential Growth and Decay Functions

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1 Setion 9. Eponential Growth and Dea Funtions Eponential Growth and Dea Funtions S Model Eponential Growth. Model Eponential Dea. Now that we an graph eponential funtions, let s learn about eponential growth and eponential dea. A quantit that grows or deas b the same perent at regular time periods is said to have eponential growth or eponential dea. There are man real-life eamples of eponential growth and dea, suh as population, bateria, viruses, and radioative substanes, just to name a few. Reall the graphs of eponential funtions. Eponential Funtions f() b f() b, for b For b For 0 b f() b, for 0 b (0, ) (, b) (0, ) (, b) Inreasing (from left to right) Eponential Growth Dereasing (from left to right) Eponential Dea Modeling Eponential Growth We begin with eponential growth, as desribed below. initial amount Eponential Growth d = C + r number of time intervals r + r is growth fator r is growth rate (often a perent) EXAMPLE In 995, let s suppose a town named Jakson had a population of 5,500 and was onsistentl inreasing b 0% per ear. If this earl inrease ontinues, predit the it s population in 05. (Round to the nearest whole.) Solution: Let s begin to understand b alulating the it s population eah ear: Time Interval = = 3 5 and so on Year Population 7,050,755 0,63,69,963 5, ,500 7, ,050 This is an eample of eponential growth, so let s use our formula with C = 5,500; r = 0.0; = = 0 Then, = C + r (Continued on net page) = 5, = 5, ,76

2 56 CHAPTER 9 Eponential and Logarithmi Funtions In 05, we predit the population of Jakson to be 0,76. Population (in thousands) (0, 0.76) (0, 5.5) Year (sine 995) In 000, the town of Jakson (from Eample ) had a population of 5,000 and started onsistentl inreasing b % per ear. If this earl inrease ontinues, predit the it s population in 05. Round to the nearest whole. Note: The eponential growth formula, = C + r, should remind ou of the ompound interest formula from the previous setion, A = P + r n nt. In fat, if the number of ompoundings per ear, n, is, the interest formula beomes A = P + r t, whih is the eponential growth formula written with different variables. Modeling Eponential Dea Now let s stud eponential dea. initial amount Eponential Dea d = C - r r number of time intervals - r is dea fator r is dea rate (often a perent) EXAMPLE A large golf ountr lub holds a singles tournament eah ear. At the start of the tournament for a partiular ear, there are 5 plaers. After eah round, half the plaers are eliminated. How man plaers remain after 6 rounds? Solution: This is an eample of eponential dea. Let s begin to understand b alulating the number of plaers after a few rounds. Round (same as interval) 3 and so on Plaers (at end of round) Here, C = 5; r = Thus, or 50, = 0.50; = 6 = = =

3 Setion 9. Eponential Growth and Dea Funtions 563 After 6 rounds, there are plaers remaining. of plaers (0, 5) (6, ) Rounds A tournament with 00 persons is plaed so that after eah round, the number of plaers dereases b 30%. Find the number of plaers after round 9. Round our answer to the nearest whole. The half-life of a substane is the amount of time it takes for half of the substane to dea. EXAMPLE 3 A form of DDT pestiide (banned in 97) has a half-life of approimatel 5 ears. If a storage unit had 00 pounds of DDT, find how muh DDT is remaining after 7 ears. Round to the nearest tenth of a pound. Solution: Here, we need to be areful beause eah time interval is 5 ears, the half-life. Time Interval 3 5 and so on Years Passed 5 # 5 = Pounds of DDT From the table, we see that after 7 ears, between and 5 intervals, there should be between.5 and 5 pounds of DDT remaining. Let s alulate, the number of time intervals. 7 ears = 5 half@life =. Now, using our eponential dea formula and the definition of half-life, for eah time interval, the dea rate r is or 50% or = : time intervals for 7 ears original amount dea rate = In 7 ears,. pounds of DDT remain. 3 Use the information from Eample 3 and alulate how muh of a 500-gram sample of DDT will remain after 5 ears. Round to the nearest tenth of a gram.

56 CHAPTER 9 Eponential and Logarithmi Funtions Voabular, Readiness & Video Chek Martin-Ga Interative Videos See Video 9. Wath the setion leture video and answer the following questions.

3. For Eample 3, whih has to do with half-life, eplain how to alulate the number of time intervals. Also, what is the dea rate for halflife and wh? 9.

4 56 CHAPTER 9 Eponential and Logarithmi Funtions Voabular, Readiness & Video Chek Martin-Ga Interative Videos See Video 9. Wath the setion leture video and answer the following questions.. Eample reviews eponential growth. Eplain how ou find the growth rate and the orret number of time intervals.. Eplain how ou know that Eample has to do with eponential dea, and not eponential growth. 3. For Eample 3, whih has to do with half-life, eplain how to alulate the number of time intervals. Also, what is the dea rate for halflife and wh? 9. Eerise Set Pratie using the eponential growth formula b ompleting the table below. Round final amounts to the nearest whole. See Eample Growth Rate per Year of Years, 305 5% 0 7% % 000 7% 9 7 9% 9 6% Final after Years of Growth Pratie using the eponential dea formula b ompleting the table below. Round final amounts to the nearest whole. See Eample Dea Rate per Year of Years, 305 5% 0 7% 5 0,000 % 5 5,000 6% 07,000 3% 5 35,000 9% 3 Final after Years of Dea MIXED Solve. Unless noted otherwise, round answers to the nearest whole. See Eamples and. 3. Suppose a it with population 500,000 has been growing at a rate of 3% per ear. If this rate ontinues, find the population of this it in ears.. Suppose a it with population 30,000 has been growing at a rate of % per ear. If this rate ontinues, find the population of this it in 0 ears. 5. The number of emploees for a ertain ompan has been dereasing eah ear b 5%. If the ompan urrentl has 60 emploees and this rate ontinues, find the number of emploees in 0 ears. 6. The number of students attending summer shool at a loal ommunit ollege has been dereasing eah ear b 7%. If 9 students urrentl attend summer shool and this rate ontinues, find the number of students attending summer shool in 5 ears. 7. National Park Servie personnel are tring to inrease the size of the bison population of Theodore Roosevelt National Park. If 60 bison urrentl live in the park, and if the population s rate of growth is.5% annuall, find how man bison there should be in 0 ears.. The size of the rat population of a wharf area grows at a rate of % monthl. If there are 00 rats in Januar, find how man rats should be epeted b net Januar. 9. A rare isotope of a nulear material is ver unstable, deaing at a rate of 5% eah seond. Find how muh isotope remains 0 seonds after 5 grams of the isotope is reated. 0. An aidental spill of 75 grams of radioative material in a loal stream has led to the presene of radioative debris deaing at a rate of % eah da. Find how muh debris still remains after das.

5 Setion 9.5 Logarithmi Funtions 565 Pratie using the eponential dea formula with half-lives b ompleting the table below. The first row has been ompleted for ou. See Eample 3. Half-Life (in ears) of Years 60 0 Years Time Intervals, = a Half@Life b Rounded to Tenths if Needed 0 Final after Time Intervals (rounded to tenths) Is Your Final Reasonable? =.5 5. es. a. 0 7 b a b Solve. Round answers to the nearest tenth. 5. A form of nikel has a half-life of 96 ears. How muh of a 30-gram sample is left after 50 ears? 6. A form of uranium has a half-life of 7 ears. How muh of a 00-gram sample is left after 500 ears? REVIEW AND PREVIEW B inspetion, find the value for that makes eah statement true. See Setions 5. and =. 3 = = 30. = 5 CONCEPT EXTENSIONS 3. An item is on sale for 0% off its original prie. If it is then marked down an additional 60%, does this mean the item is free? Disuss wh or wh not. 3. Uranium U-3 has a half-life of 7 ears. What eventuall happens to a 0 gram sample? Does it ever ompletel dea and disappear? Disuss wh or wh not. 9.5 Logarithmi Funtions S Write Eponential Equations with Logarithmi Notation and Write Logarithmi Equations with Eponential Notation. Solve Logarithmi Equations b Using Eponential Notation. 3 Identif and Graph Logarithmi Funtions. Using Logarithmi Notation Sine the eponential funtion f = is a one-to-one funtion, it has an inverse. We an reate a table of values for f - b swithing the oordinates in the aompaning table of values for f = f 0 3 f f() 3 5 f ()

Check Skills You ll Need (For help, go to Lesson 1-2.) Evaluate each expression for the given value of x.

Check Skills You ll Need (For help, go to Lesson 1-2.) Evaluate each expression for the given value of x. A_3eSE_00X 0/6/005 :3 AM Page - Eploring Eponential Models Lesson Preview What You ll Learn To model eponential growth To model eponential deca... And Wh To model a car s depreciation, as in Eample 6 Check