求解决高中数学问题3.(a)Considerarationalfunctionp(x)withverticalasymptotesatx=3andx=−3.Thegraphofp(x)showsahorizontalasymptoteaty=4,andayinterceptatthepoint(0,−1).(i)Fromtheinformationpr

求解决高中数学问题3.(a)Considerarationalfunctionp(x)with

verticalasymptotesatx=3andx=−3.Thegraphofp(x)showsahorizontal

asymptoteaty=4,andayinterceptatthepoint(0,−1).

(i)Fromtheinformationprovided,writethefunctionp(x)(4)

(ii)UsingAutographorotherwise,drawa

graphofthefunctionp(x)includingand
labelingitsverticalandhorizontal

asymptotesaswellasanyintercepts.(3)

(b)Atruckisona500kmjourney.Itistraveling

atafairlyconstantspeedonareasonablystraightstraightandlevelroadand

burnsdieselfuelatarateofg(x)litresperkilometre,

800+x2wherexistheaveragespeedofthetruckinkilometresperhour,andg(x)=200x.

2980+3.725x2(i)Thecostofdieselis$1.49perlitre.ShowthatC(x)=x

thefuelcostforthistrucktravelingatanaveragespeedofxkm/h.

whereCisthefuelcostforthistrucktravelingatanaveragespeedofxkm/h.

(ii)UsingAutograph,orotherwise,drawa

graphofC(x)withayaxisrangingfrom0to500,andanxaxisrangingfrom0

to80.Interpretandlabeleachaxis.(3)

(iii)Fromyourgraph,findtherangeof

averagedrivingspeeds,tothenearestwholenumber,thatwillkeepthefuel

costsCx)forthe500kmtripto$300orless.(2)

(iv)Fromyourgraph,estimatetothenearest

wholenumber,thedrivingspeedthatwillminimisethefuelcostforthetrip.

Commentonitspracticality,andanythingthatmightinvalidatethisspeedas

indicatingminimumfuelusage.(3)

(v)Verifyyouranswerinpart(c)algebraicallybysolvingtheinequality,C(x)≤300.(2)

4.Aresearcherisinterestedinthebuying

behaviour(anindicatorofdemand)ofaselectedgroupofpartypillusers.In

acontrolledexperiment,anewlegalisedpartypillisputontothemarket.

Basedonpreviousdata,theresearchersuggeststhefollowingdemandmodel:

D(t)=100−95(3/5)ttismeasuredinthenumberofweeksithasbeenonthe

market,andD(t)representsthepercentageoftheselectedgroupwantingtobuy

thepartypillaf