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IntroductiontoExponentialFunctionsExerciseSet5
1. Foranexperiment,ascientistdesignsacan,20cminheight,thatcanholdwater.Atubeisinstalledatthebottomofthecanallowingwatertodrainout.
Atthebeginningoftheexperiment,thecanisfull.Whentheexperimentstarts,thewaterbeginstodrain,andtheheightofthewaterinthecandecreasesbyafactorof!
!
eachminute.
a. Explainwhytheheightofthewaterinthecanisafunctionoftime.
b. Theheight,ℎ,incm,isafunction𝑓oftime𝑡inminutessincepokingthehole,ℎ = 𝑓(𝑡).Findanexpressionfor𝑓(𝑡).
c. Findandrecordthevaluesfor𝑓when𝑡is0,1,2,and3.
d. Find𝑓(4).Whatdoes𝑓(4)represent?
e. Sketchagraphof𝑓byhandoruseDesmos.
f. Whathappenstothelevelofwaterinthecanastimecontinuestoelapse?Howdoyouseethisinthegraph?
2. Ascientistmeasurestheheight,ℎ,ofatreeeachmonth,and𝑚isthenumberofmonthssincethescientistfirstmeasuredtheheightofthetree.
a. Istheheight,ℎ,afunctionofthemonth,𝑚?Explainhowyouknow.
b. Isthemonth,𝑚,afunctionoftheheight,ℎ?Explainhowyouknow.
3. Abacteriapopulationis10,000.Ittripleseachday.
a. Explainwhythebacteriapopulation,𝑏,isafunctionofthenumberofdays,𝑑,sinceitwasmeasuredtobe10,000.
b. Whichvariableistheindependentvariableinthissituation?
c. Writeanequationrelating𝑏and𝑑.
a. Istheposition,𝑝,oftheminutehandonaclockafunctionofthetime,𝑡?
b. Isthetime,𝑡,afunctionofthepositionoftheminutehandonaclock?
4. Theareacoveredbyacityis20squaremiles.Theareagrowsbyafactorof1.1eachyearsinceitwas20squaremiles.
a. Explainwhythearea,𝑎,coveredbythecity,insquaremiles,isafunctionof𝑡,thenumberofyearssinceitsareawas20squaremiles.
b. Writeanequationfor𝑎intermsof𝑡.
5. Thenumberofpeoplewiththefluduringanepidemicisafunction,𝑓,ofthenumberof
days,𝑑,sincetheepidemicbegan.Theequation𝑓(𝑑) = 50 ⋅ !!
!defines𝑓.
a. Howmanypeoplehadthefluatthebeginningoftheepidemic?Explainhowyouknow.
b. Howquicklyisthefluspreading?Explainhowyoucantellfromtheequation.
c. Whatdoes𝑓(1)meaninthissituation?
d. Does𝑓(3.5)makesenseinthissituation?
6. Thefunction,𝑓,givesthedollarvalueofabond𝑡yearsafterthebondwaspurchased.Thegraphof𝑓isshown.
a. Whatis𝑓(0)?Whatdoesitmeaninthissituation?
b. Whatis𝑓(4.5)?Whatdoesitmeaninthissituation?
c. Whenis𝑓(𝑡) = 1500?Whatdoesthismeaninthissituation?
7. Technologyrequired.Afunction𝑓givesthenumberofstraycatsinatown𝑡yearssincethetownstartedananimalcontrolprogram.Theprogramincludesbothsterilizingstraycatsandfindinghomestoadoptthem.Anequationrepresenting𝑓is𝑓(𝑡) =243 !
!
!.
a. Whatisthevalueof𝑓(𝑡)when𝑡is0?Explainwhatthisvaluemeansinthissituation.
b. Whatistheapproximatevalueof𝑓(𝑡)when𝑡is!!?Explainwhatthisvalue
meansinthissituation.
c. Whatdoesthenumber!!tellyouaboutthestraycatpopulation?
d. Usetechnologytograph𝑓forvaluesof𝑡between0and4.Whatgraphingwindowallowsyoutoseevaluesof𝑓(𝑡)thatcorrespondtothesevaluesof𝑡?
8. Function𝑔givestheamountofachemicalinaperson'sbody,inmilligrams,𝑡hours
sincethepatienttookthedrug.Theequation𝑔(𝑡) = 600 ⋅ !!
!definesthisfunction.
a. Whatdoesthefraction!!meaninthissituation?
b. Sketchagraphof𝑔.
c. Whatarethedomainandrangeof𝑔?Explainwhattheymeaninthissituation.
9. Thedollarvalueofamopedisafunctionofthenumberofyears,𝑡,sincethemoped
waspurchased.Thefunction,𝑓,isdefinedbytheequation𝑓(𝑡) = 2, 500 ⋅ !!
!.
Whatisthebestchoiceofdomainforthefunction𝑓?
a. -10 ≤ 𝑡 ≤ 10
b. -10 ≤ 𝑡 ≤ 0
c. 0 ≤ 𝑡 ≤ 10
d. 0 ≤ 𝑡 ≤ 100
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