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