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Writeyourquestionshere!
8.4SolvingExponentialEquationsSolvetheequationgraphically….PARTII!!!!Ex1: Ex2:What’stheinverseofanexponentialfunction? Ex3:LogarithmProperty
FindtheEXACTandAPPROXIMATEanswersEx4: Ex5:Youtry!Useotherpropertiestosolve!Ex6: Ex7:
NOTES
©2018TheAlgebros,LLC
Mr.Kellywasgiven$500forhishighschoolin1995.Heputsinabankthatpromisestheinvestmentwillgrowby12%eachyear.Howmanyyearswillheneedtoinvestifhewaitstillhehas$5000inhisaccount?Howmuchquickerwoulditbeifhefoundabanktocompoundthatsame$5000,at12%continuously?
YOUTRY!!!!!Ex1: Ex2:
SUMMARY:
CompoundingInterest(continuouscompounding)
CompoundingInterest(periodiccompounding)
%increase/decreaseperunitoftime
𝐴 = 𝑃𝑒!" 𝐴 = 𝑃 1 +𝑟𝑛
!" 𝑓(𝑥) = 𝑎𝑏!
©2018TheAlgebros,LLC
8.4SolvingExponentialEquations
Directions:Solvetheequationwithagraphingcalculator.Roundtothenearestthousandth.1)4!! = 12!!! 2)2𝑒! = 4!!! 3)4 3! = 5 !
!
!!!
Directions:Solvetheequation.GivetheEXACTandAPPROXIMATE(tonearestthousandth)answers.4)5!!! = 24 5)74 = 8!!
6)8!!! − 5 = 53 7)−6𝑒!!! − 4 = −24
8)2 3!!! − 10 = 32 9)2 4! = 4!!
PRACTICE
©2018TheAlgebros,LLC
10)15 2!!! = 3 2!! 11)4𝑒!!!! = 20𝑒!
CompoundingInterest(continuouscompounding)
CompoundingInterest(periodiccompounding)
%increase/decreaseperunitoftime
𝑨 = 𝑷𝒆𝒓𝒕 𝑨 = 𝑷 𝟏 +𝒓𝒏
𝒏𝒕 𝒇(𝒙) = 𝒂𝒃𝒙
12)Mr.Brustinvests$5000inanaccountthatiscompoundedmonthlyatarateof9%.Howmanyyearswillittakehimtohave$23,000intheaccount?
13)Mr.Beanputs$5000inamutualfundthatincreasesinvalueby11%eachyear.Howmanyyearswillittakehimtoreach$23,000inhismutualfund?
14)Mr.Kellyputs$5000intoanaccountthatiscompoundedcontinuouslyatarateof10%.Howlongwillittakeforhimtohave$23,000intheaccount?
15)AthisweddingMr.KellyhadastatueoftheAlgebrosmadeofice.Heestimatedthatevery4dayshalfoftheweightoftheicesculpturehasmelted.Howmanydaysuntil10%ofthesculptureisleft?16)Mr.Beancollectscrickets.Hehastencricketscurrentlyandrealizesthepopulationistriplingevery2weeks.Howlongwillittakeforhiscricketpopulationtoreach500?
©2018TheAlgebros,LLC
8.4SolvingExponentialEquations
Directions:Solvetheequation.GivetheEXACTandAPPROXIMATE(tonearestthousandth)answers.
2)Mr.SullivanstartsupanewcompanycalledSullivan’sSell-a-Van.Hebelieveshisinitialinvestmentof$15,000willincreaseyearlybyarateof7%.Howlongtillhiscompanyisworth$50,000?
1)2 5!!! = 8 5!!
3)Mr.BrustisintalkswithaprinceinNigeriaviaemail.TheprincetellsMr.Brusthe’llgivehimachoice.Hecaninvest$1000withtheprince’sconstructioncompanyinwhichtheprinceguaranteeshe’lldoublehismoneyevery5weeks.Orhecaninvestthesamemoneyintohistelecombusinessinwhichtheprincesayshe’lltriplehismoneyevery8weeks.a)Withoutdoinganymath,whichcompanydoyouthinkwouldbethebestchoiceforMr.Brustifhewantstomakethemostmoneyasquicklyaspossible?Explainyourreasoning?b)IfTHISWEREACTUALLYreal,whichcompanyshouldMr.Brustinvestinifhewantstomake$200,000thefastest?Howmuchfasterisit?4)Thepopulationofacertaincountrywas40millionintheyear2000andhasgrowncontinuallyintheyearsfollowing.ThepopulationP,inmillions,ofthecountrytyearsafter2000canbemodeledbythefunction𝑃 𝑡 = 40𝑒!.!"#!,where𝑡 ≥ 0.PARTABasedonthemodel,thesolutiontotheequation50 = 40𝑒!.!"#!givesthenumberofyearsitwilltakeforthepopulationofthecountrytoreach50million.Whatisthesolutiontotheequationexpressedasalogarithm.A.0.027 ln 1.25
B.!" (!.!")!.!"#
C.ln !.!"!.!"#
D.ln !.!"#!.!"
WRAP UP
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PARTBWhichwouldaffecthowlongittakesforthepopulationtoreach50millionfaster.Option1:Youalteredtheinitialpopulation(raiseditby5millionpeople).OPTION2:Youdoubletheratethepopulationincreased.EXPLAINyourreasoning!PARTCWhichoptionwouldbefastestifyouwantedtoknowhowlongitwouldtakeforthepopulationtoreach60million.EXIT TICKET –
CompoundingInterest(periodiccompounding)
𝑨 = 𝑷 𝟏 +𝒓𝒏
𝒏𝒕
After school, you will undoubtedly want to save money. If you are able to find several banks that all offer compound interest for your account, then what other variable would be most important to you when you are comparing the different banks? Explain your answer!
