Whenelectricitybecamewidelydistributedduringtheearlypartofthe 20Th Century

Whenelectricitybecamewidelydistributedduringtheearlypartofthe 20th century, stategovernmentsregulatedtheelectricityindustryasamonopoly.Oneelectriccompanyhadthe rightstogenerateanddistributeelectricityforacityoracertain part ofthe state.Inreturn,thegovernmentlaidoutasetofrulesforwhattheelectriccompanycouldandcould notdo.

Duringthe1990sandearly 21stcentury, many states deregulated electricity. As a result,numerouselectric companiescan now provideelectricity for a particular area. OnesuchcompanyisLightsandPower.Toattractcustomers,LightsandPowerisadvertisinga special:

1.Accordingtotheadvertisement,howmuchdoesthefirst1,000kilowatt-hours (kWh)ofelectricitycost acustomer?

2.SupposeMrs.Brownuses1,200kilowatt-hoursofelectricity. Howmuchdoesshepayforthefirst1,000kilowatt-hours?

Howmuchdoesshepayforthenext200 kilowatt-hours of electricity?Howmuchdoesshepayaltogetherfor1,200kilowatt-hoursof electricity?

3.UsetheinformationinLightsandPower’sadvertisementtodeterminethecostofelectricity for the amounts listed in the table.

AmountofElectricity(kWh) / Process / Cost($)
700 / 700(0.11) / 77
800
900
1,000
1,100
1,200 / 1,000(0.11)+ (1,200– 1,000)(0.18) / 146
1,300
1,400
1,500
1,600
1,700
1,800
1,900

4.Writean equationtodescribe thecost(y) ofthe numberofkilowatt-hours of electricity

(x)to1,000kilowatt-hours.

5.Forwhat domain does your function model thecost ofthefirst 1,000 kilowatt-hours ofelectricity?

6.Writean equationto describe the cost(c) of thenumber of kilowatt-hours ofelectricity

(x)from1,001 to 1,500 kilowatt-hours.

7.Forwhat domain does your functionmodel the cost of 1,001 to1,500 kilowatt-hours ofelectricity?

8.Writean equationto describe the cost (m) ofthe numberofkilowatt-hours ofelectricity

(x)morethan1,500kilowatt-hours.

9.Forwhat domain doesyour function modelthe cost ofmorethan 1,500 kilowatt-hoursofelectricity?

10.Writethreepiecewisefunctions,includinglimitationsonthedomain,thatdescribethe costofpurchasingelectricityfromLightsandPower.

11.Useyourgraphingcalculatortomakeascatterplotofcostversusamountofelectricity.Describetheaxesandscalingandsketchyourgraph.

12.Graphyourpiecewisefunctionsoveryour scatterplot. Usethedomainrestrictions.Howwell dothe functions model thedatageneratedbythe electricityplan?

13.Thefunction y= 0.11x hasadomain of all real numbers.Whyisthe domain of thefunctionasitis applied inthis situation restricted?

AsofMay2009,theU.S.PostalServiceadjusteditsratessothatmailingalargeenvelope costs$0.88 for the first ounceand $0.17foreachadditionalounce.Thereisaweightlimitforallfirst-classmail—lettersandparcelsmailedfirstclasscannotexceed13ounces.

Considerthegraphbelow.

Weight(Ounces)

14.Whattypeoffunctionisrepresentedbythegraph?Howdoyou know?

15.IsthistypeoffunctionappropriatetorepresenttheU.S.PostalServiceratesforsendinglargeenvelopesby first-classmail?Whyor why not?

16.HowwelldoesthegraphrepresenttheU.S.PostalServiceratesforsendinglarge envelopesby first-classmail? Howdoyouknow?

17.Howcouldyoumodifythegraphtobetterrepresentthesituation?

18.REFLECTION:Whattypes of situationscana step function beused to model?

19.REFLECTION:How arestep functionssimilar to piecewisefunctions? Howare theydifferent?

20.EXTENSION:Researchtaxicabfaresforyourcityoracitythatyouwanttovisit.Whattypeoffunctionismostappropriatetorepresentthosefares?Generateagraphtoshow thefaresandpresentyourfindingstothe class.

21.EXTENSION:Researchtodetermineanappropriateresponsetothefollowingquestion.Prepareashortpresentationofyourfindings.

Wouldfederalincometaxesbebettermodeledwithastepfunctionorapiecewise function?