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