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Mathematics

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Mathematics1MathematicsDepartmentPhillipsExeterAcademyExeter,NHAugust2012TotheStudentContents: Members of the PEAMathematics Department have writtenthe materialinthis book. As youworkthroughit, youwill discover that algebra, geometry, andtrigonometry have been integrated into a mathematical whole. There is no Chapter 5, noristhereasectionontangentstocircles. Thecurriculumisproblem-centered, ratherthantopic-centered. Techniquesandtheoremswill becomeapparentasyouworkthroughtheproblems, andyouwill needtokeepappropriatenotesforyourrecordstherearenoboxes containingimportant theorems. There is noindex as such,butthe referencesectionthatstartsonpage201shouldhelpyourecallthemeaningsofkeywordsthataredenedintheproblems(wheretheyusuallyappearitalicized).Commentsonproblem-solving: You should approach each problem as an exploration.Readingeachquestioncarefullyis essential, especiallysince denitions, highlightedinitalics, areroutinelyinsertedintotheproblemtexts. Itisimportanttomakeaccuratediagramswheneverappropriate. Useful strategiestokeepinmindare: createaneasierproblem, guessandcheck, workbackwards, andrecallasimilarproblem. Itisimportantthatyouworkoneachproblemwhenassigned,sincethequestionsyoumayhaveaboutaproblemwilllikelymotivateclassdiscussionthenextday.Problem-solving requires persistence as much as it requires ingenuity. When you get stuck,or solve a problem incorrectly, backup and start over. Keepin mind thatyoure probablynottheonlyonewhoisstuck,andthatmayevenincludeyourteacher. Ifyouhavetakenthetimetothinkabout aproblem, youshouldbringtoclass awrittenrecordof youreorts, notjustablankspaceinyournotebook. Themethodsthatyouusetosolveaproblem, thecorrectionsthatyoumakeinyourapproach, themeansbywhichyoutestthe validity of your solutions,and your ability to communicate ideas are just as importantasgettingthecorrectanswer.About technology: Manyof theproblemsinthisbookrequiretheuseof technology(graphingcalculatorsorcomputer software) inordertosolvethem. Moreover, youareencouragedtousetechnologytoexplore, andtoformulateandtest conjectures. Keepthefollowingguidelinesinmind: writebeforeyoucalculate,sothatyouwillhaveaclearrecord of what you have done; store intermediate answers in your calculator for later use inyour solution;payattention tothedegreeof accuracyrequested;referto yourcalculatorsmanualwhenneeded; andbepreparedtoexplainyourmethodtoyourclassmates. Also,if youareaskedtography=(2x 3)/(x + 1), forinstance, theexpectationisthat,althoughyoumight useyour calculator togenerateapictureof thecurve, youshouldsketchthatpictureinyournotebookorontheboard,withcorrectlyscaledaxes.PhillipsExeterAcademyIntroductoryMathGuideforNewStudents(Forstudents,bystudents!)IntroductionAnnually, approximately 300newstudentstake upstudiesin theMathematics Depart-ment. Comingfromvariousstylesofteaching,asanewstudentyouwillquicklycometorealizethedistinctmethodsandphilosophiesofteachingatExeter. OneaspectofExeterthatoftencatchesstudentsunawareisthe mathcurriculum. Iencourageallnewstudentsto come to the math table with a clear mind. You may not grasp, understand,or even likemathatrst,butyouwillhavetobepreparedforanythingthatcomesbeforeyou.Duringthe fall of 2000, the newstudents avidlyvoicedaconcernabout the mathcurriculum. Our concernrangedfromgrading, tomathpolicies, andeventotheverydierentteachingstyles utilized in the mathematics department. The guide that you havebegun reading was written solely by students, with the intent of preparing you for the taskthatyouhaveembarkedupon. Thisguideincludestipsforsurvival, testimonialsofhowwefeltwhenenteringthemathclassroom,andaspectsofmaththatwewouldhavelikedto have known,before we feltoverwhelmed. Hopefully, this guide will easeyour transitionintomath atExeter. Remember,Anythingworth doing,is hardto do. Mr. Higgins 36.AnthonyL.Riley04Ilearnedalotmorebyteachingmyselfthanbybeingtaughtbysomeoneelse.Onelearnsmanywaystododierentproblems. Sinceeachproblemisdierent,youareforcedtouseall aspectsofmath.Ittakeslongerfornewconceptstosinkin. . . youunderstand,butbecauseitdidntsinkin,itsveryhardtoexpandwiththatconcept.Itmakesmethinkmore. Thewaythemathbooksaresetup(i.e. simpleproblemsprogressingtoharderonesonaconcept)reallyhelpsmeunderstandthemathematical concepts.Whenyoudiscoverorformulateaconceptyourself,yourememberitbetterandunderstandtheconceptbetterthanifwememorizeditortheteacherjusttoldusthattheformulawasxyz.HomeworkMathhomework=noexplanationsandeightproblemsanight. Forthemostpart, ithasbecomestandardamongmostmathteacherstogiveabouteight problemsanight;butIhaveevenhadateacherwhogavetenthoughtwoproblemsmaynotseemlikeabigdeal, it canbe. Sinceall theproblems arescenarios, andoftenhavetopics thatvary,theyalsorangein complexity,from asimple,one-sentencequestion,toafull-edgedparagraph with an eight-part answer! Dont fret though, transition to homework will comewithtime,similartohowyougainwisdom,asyougetolder. Homeworkcanvarygreatlyfromnighttonight, sobeexiblewithyourtimethisleadstoanotherpartof doingyour homework. IN ALL CLASSES THAT MEET FIVE TIMES A WEEK, INCLUDINGMATHEMATICS,YOUSHOULDSPEND50MINUTES ATTHEMAXIMUM,DOINGHOMEWORK! Noteacher shouldever expect youtospendmoretime, withthelargeworkloadExonianscarry. Tryyourhardesttoconcentrate, andutilizethose50minutesasmuchaspossible.iWithoutanyexplanations showingyou exactly howtodoyour homework,howare yousupposedtodoaproblemthatyouhaveabsolutelynoclueabout?(ThisWILLhappen!)Asksomebodyinyourdorm. Anotherpersoninyourdormmightbeinthesameclass,orthesamelevel, anditisalwayshelpful toseektheassistanceof someoneinahigherlevel of math. Also remember,there is a dierencebetweenhomeworkand studying;afteryourethroughwiththeeightproblemsassignedtoyou,gobackoveryourworkfromthelastfewdays.. . . withhomework,youwouldntgetmarkeddownifyoudidntdoaproblem.GoingtotheBoardItisveryimportanttogototheboardtoputuphomeworkproblems. Usually,everyhomeworkproblemisputupontheboardatthebeginningof class, andthentheyarediscussedinclass. Ifyouregularlyputproblemsupontheboard,yourteacherwillhaveagoodfeel ofwhereyoustandintheclass; acondentstudentwill mostlikelybemoreactiveinparticipatingintheclass.PlagiarismOne thingtokeepinmindis plagiarism. Youcanget helpfromalmost anywhere,but make sure that youcite your help, andthat all workshownor turnedinis yourown, evenifsomeoneelseshowedyouhowtodoit. Teachersdooccasionallygiveprob-lems/quizzes/teststobecompletedathome. Youmaynotreceivehelpontheseassess-ments,unlessinstructedtobyyourteacher;itisimperativethatalltheworkisyours.MathExtra-HelpGettinghelpisanintegralpartofstayingontopofthemathprogramhereatExeter.Itcanberatherfrustratingtobelostandfeelyouhavenowheretoturn. Thereareafewtricksofthetradehowever, whichensureyoursafety,withthispossiblyoverwhelmingwordproblemextravaganza.TeachersandMeetingsTheveryrstplacetoturnforhelpshouldbeyourteacher. SinceteachersatExeterhave many fewer studentsthan teachers at other schools,they are never less than eager tohelp you succeed in any way they can. There is actually one designated time slot a week forstudents to meet with teachers,which is meetings period on Saturday. You can always calloraskateacherforhelp. Ifthereisnotimeduringtheday,itisalwayspossibletocheckout of the dorm after your check-in time, to meetwith your teacher at their apartment, orhouse. Itiseasiesttodothisonthenightsthatyourteacherisondutyinhis/herdorm.Gettinghelpfromyourteacheristherstandmostreliablesourcetoturnto, forextrahelp.Youcouldmeetwiththeteacherforextrahelpanytime.Extrahelpsessionsone-on-onewiththeteacher. Myoldmathtext.ii7-9MathHelpAlong with helpfrom your teacher,thereare severalother placestogethelp. From 7-9PMeverynight, exceptSaturday, thereisaMathandSciencehelpgroupintheScienceCenter. Each evening, the lab is lled with students in a broad range of math levels, whichshouldbeabletohelpyouwithproblemsyouhave. Also,rememberthatyourhomeworkis notgradedeveryday,andyour teacherwill usuallytellyou whenhe/shewillbe gradingaparticular assignment. Thismeansthatyoucanalwaysndsomeoneinyourdormthatwill help you catchupor simplyhelp you with atough problem. If you are adaystudent,IwoulddenitelyrecommendgoingtoScienceandMathHelp.. . . hardertounderstandconceptsifyoudontunderstandaproblembecauseeachproblemistryingtoteachyousomethingdierentthatleadstoanewconcept.Hardtoseparatedierentmathconcepts. NotsurewhatkindofmathitisImlearning.Morediculttoreview.DierentTeachersTeachDierentlyTheteachersatExeterusuallydeveloptheirownstyleofteaching, ttedtotheirphi-losophy of the subject they teach;it is no dierent in the math department. Teachers varyat all levels: they grade dierently, assess your knowledge dierently, teach dierently, andgooverhomeworkdierently. Theyoerhelpdierently, too. Thissimplymeansthatitisessential thatyoubepreparedeachtermtoadapttoaparticularteachingstyle. Forinstance,my teacher tests me about every two weeks,gives hand-in problems every coupleofdays,andalsogivesafewquizzes. However, myfriend, whoisinthesamelevelmathasIam,hasateacherwhodoesntgiveanytestsorquizzes;heonlygradesonclasspar-ticipation,andassignsasinglehand-inproblem,eachassignment. Dontbeafraidtoaskyourteacherhowtheygrade, becausethiscanbecomeverycrucial; variousteachersputmore weight on class participation in grading while others do the opposite. You must learnto be exible to teachingstyles and even your teachers personality. This is a necessityforalldepartmentsatExeter,includingmath.Thetestsarethehardestpartbetweentermstoadaptto,butifyoupreparewell,thereshouldntbeaproblem.Testsarehard. Cantgoatyourownpace.Myotherteachertaughtandpointedoutwhichproblemsarerelatedwhentheyaresixpagesapart.Ittookafewdaysadjustingto,butifyoupayattentiontowhattheteachersaysandaskhim/herquestionsabouttheirexpectations,transitionsshouldbesmooth.Inconsistent. Everyteachergavedierentamountsofhomeworkandtests. Classworkvariedtoo. Myfall termteachermadeusputeveryproblemontheboard,whereasmywintertermteacheronlyconcentratedonafew.JonathanBarbee04RyanLevihn-Coon04iiiNewStudentTestimonialsTherewasnotafoundationtobuildon. Therewerenoexampleproblems.After eightyears of math textbooks and lecture-stylemath classes,math at Exeter wasa lot to get used to. My entire elementary math education was based on reading how to doproblemsfromthetextbook, thenpracticingmonotonousproblemsthathadnoreal-liferelevance, oneaftertheother. Thismethodisneforsomepeople,butitwasntforme.BythetimeIcametoExeter,Iwasreadyforachangeofpace,andIcertainlygotone.Having somewhat of a background in algebra, I thought the Transition 1 course was justrightforme. Itwentoverbasicalgebraandproblem-solvingtechniques. ThemathbooksatExeterareverydierentfromtraditional books. Theyarecompiledbytheteachers,and consist of pages upon pages of word problems that lead you to nd your own methodsof solvingproblems. Theproblemsarenotveryinstructional, theylaytheinformationdown for you, most times introducing new vocabulary, (there is an index in the back of thebook), and allow you to think about the problem, and solve it any way that you can. WhenIrstusedthisbooklet, Iwasalittlethrownback; itwassodierentfromeverythingIhaddonebeforebutbythetimethetermwasover,Ihadthenewmethoddown.Theactual mathclassesatExeterwerehardtogetusedtoaswell. Teachersusuallyassignabouteightproblemsanight,leavingyoutimetoexploretheproblemsandgiveeachonesomethought. Then, next class, students put all thehomeworkproblems ontheboard. The class goes over eachproblem; everyoneshares their methodandevendicultiesthattheyranintowhilesolvingit. Ithinkthehardestthingtogetusedto,isbeingabletoopenlyaskquestions. Noonewants tobewrong, I guess it ishumannature, but in the world of Exeter math, you cant be afraid to ask questions. You have toseize the opportunity to speak up and say I dont understand,or How did you get thatanswer?Ifyoudontaskquestions,youwillnevergettheanswersyouneedtothrive.Something that my current math teacher always says is to make all your mistakes on theboard, because when a test comes around, you dont want to make mistakes on paper. Thisissotrue,classtimeispracticetime,anditshardtogetusedtonotfeelingembarrassedafteryou answerproblemsincorrectly. You needtogooutonalimbandtryyourbest. Ifyou get a problem wrong on the board, its one new thing learned in class, not to mention,onelessthingtoworryaboutmessingupon,onthenexttest.MathatExeterisreallybasedoncooperation,you,yourclassmates,andyourteacher.Ittakesawhiletogetusedto,butintheend,itisworththeeort.HazelCipolle04ivAtrst,Iwasveryshyandhadahardtimeaskingquestions.Sometimesotherstudentsdidntexplainproblemsclearly.Solutionstocertainproblemsbyotherstudentsaresometimesnotthefastestoreasiest.Somestudentsmightknowtricksandspecialtechniquesthatarentcovered.IenteredmysecondmathclassofFallTermasaninthgrader,withafeelingofdread.Though I had understood the homework the night before, I looked down at my paper witha blank mind, unsure how I had done any of the problems. The class sat nervously aroundthetableuntilwewerepromptedbytheteachertoputthehomeworkontheboard. Oneboystoodupandpickedupsomechalk. Soonothersfollowedsuit. Istayedgluedtomyseatwiththesamequestionrunningthroughmymind,whatifIgetitwrong?Iwasconvincedthateveryonewouldmakefunof me, thattheywouldtearmyworkapart, that each person around that table was smarter than I was. I soon found that I wastheonlyonestillseatedandhurriedtotheboard. TheonlyavailableproblemwasoneIwasslightlyunsureof. Iwrotemyworkquicklyandreclaimedmyseat.Wereviewedthedierentproblems,andeveryonewassuccessful. Iexplainedmyworkandawaitedtheclassresponse. Myclassmatesagreedwiththebulkofmywork,thoughtherewasaquestionononepart. Theysuggesteddierentwaystondtheanswerandwewereabletoworkthroughtheproblem,together.I returnedtomy seatfeelingmuchmore condent. Not only weremy questions clearedup,butmyclassmatesquestionswereansweredaswell. Everyonebeneted.IlearnedoneofthemoreimportantlessonsaboutmathatExeterthatday;itdoesntmatterifyouarerightorwrong. Yourclassmateswillbesupportiveofyou,andtolerantof yourquestions. Chancesare, if youhadtroublewithaproblem, someoneelseintheclassdidtoo. Anotherthingtokeepinmindisthattheteacherexpectsnothingmorethanthatyoutrytodoaproblemtothebestof yourability. If youexplainaproblemthatturnsouttobeincorrect, theteacherwill notjudgeyouharshly. Theyunderstandthatnooneisalwayscorrect,andwillnotbeangryorupsetwithyou.ElisabethRamsey04vMybackgroundinmathwasalittleweakerthanmostpeoples,thereforeIwasunsurehowtodomanyoftheproblems. IneverthoroughlyunderstoodhowtodoaproblembeforeIsawitinthebook.Ineverthoughtmathwouldbeaproblem. Thatis, until IcametoExeter. IenteredintoMathT1B,cluelessastowhatthecurriculumwouldbe. ThedayIboughttheMathOnebookfromtheBookstoreAnnex, Istaredattheproblemsindisbelief. ALLWORDPROBLEMS. Why word problems? I thought. I had dreaded word problems ever since Iwas a second grader, and on my comments it always read, Charly is a good math student,butsheneedstoworkonwordproblems.Iwasinshock. Iwouldhavetolearnmathinanentirelynewlanguage. IbegantodreadmyBformatmathclass.Myrst mathtest at Exeter was horrible. I hadnever seenaDonamathtest.Never. IwasupsetandIfeltdumb, especiallysinceothersinmyclassgotbettergrades,andbecausemyroommatewasextremelygoodinmath. Icried. IsaidIwantedtogohomewherethingswereeasier. ButnallyIrealized, Iwasbeinggivenachallenge. Ihadtoatleasttry.Iwenttomymathteacherforextrahelp. Iaskedquestionsmoreoften(thoughnotasmuch as I should have), and slowly I began to understand the problems better. My gradesgraduallygotbetter,bygoingfromaDtoaC+toaBandeventuallyIgotanA. Itwashard, butthatisExeter. Youjusthavetogetpassedthatrsthump, thoughlittleoneswill follow. Aslongasyoudontcompareyourself toothers, andyouaskforhelpwhen you need it, you should get used to the math curriculum. I still struggle, but as longasIdontgetintimidatedanddontgiveup,Iamabletobringmygradesup.CharlySimpson04Theabovequotesinitalicsweretakenfromasurveyofnewstudentsinthespringof2001.viMathematics11. Lighttravels at about 186 thousand miles per second,and the Sun is about 93 millionmilesfromtheEarth. HowmuchtimedoeslighttaketoreachtheEarthfromtheSun?2. Howlongwouldittakeyoutocount toonebillion, recitingthenumbersoneafteranother? First writeaguess intoyour notebook, thencomeupwithathoughtful an-swer. Oneapproachistoactuallydoitandhavesomeonetimeyou, buttherearemoremanageablealternatives. Whatassumptionsdidyoumakeinyourcalculations?3. Ittakes 1.25secondsforlighttotravelfromtheMoon totheEarth. How manymilesawayistheMoon?4. Manymajor-league baseballpitcherscanthrow theball at90miles perhour. Atthatspeed, howlongdoesittakeapitchtotravel fromthepitchersmoundtohomeplate,adistanceof 60feet6inches? Giveyouranswertothenearesthundredthof asecond.Thereare5280feetinamile.5. You have perhaps heard the saying, A journey of 1000 miles begins with a single step.Howmanystepswouldyoutaketonishajourneyof1000miles? Whatinformationdoyouneedinordertoanswerthisquestion? Findareasonableanswer. Whatwouldyouranswerbeifthejourneywere1000kilometers?6. In an oshore pipeline, a cylindrical mechanism called a pig is run through the pipesperiodicallytocleanthem. Thesepigstravel at2feetpersecond. Whatisthisspeed,expressedinmilesperhour?7. Yourclasssponsorsabenetconcertandpricestheticketsat$8each. Dalesells12tickets,Andy16,Morgan17,andPat13. Computethetotalrevenuebroughtinbythesefourpersons. Noticethattherearetwowaystodothecalculation.8. Kelly telephoned Brook about a homework problem. Kelly said, Four plus three timestwois14, isntit?Brookreplied, No, its10.Didsomeonemakeamistake? Canyouexplainwherethesetwoanswerscamefrom?9. It is customaryinalgebratoomit multiplicationsymbols whenever possible. Forexample,11x meansthe samethingas 11 x. Whichof thefollowing canbe condensedbyleavingoutamultiplicationsymbol?(a)4 13(b)1.08 p (c)24 52 (d)5 (2 + x)10. WesboughtsomeschoolsuppliesatanoutletstoreinMaine,astatethathasa6.5%salestax. Includingthesalestax,howmuchdidWespayfortwoblazerspricedat$49.95eachand3pairsofpantspricedat$17.50each?11. (Continuation)Afamiliarfeatureofarithmeticisthatmultiplicationdistributesoveraddition. Written in algebraic code,this property looks like a(b +c) = ab +ac. Because ofthisproperty, therearetwoequivalentmethodsthatcanbeusedtocomputetheanswerinthepreviousproblem. Explain,usingwordsandcompletesentences.August2012 1 PhillipsExeterAcademyMathematics11. Woolworths had a going-out-of-businesssale. The price of a telephonebefore the salewas $39.98. Whatwas the priceof the telephoneafter a 30% discount?If thesale priceofthesametelephonehadbeen$23.99,whatwouldthe(percentage)discounthavebeen?2. Pickanynumber. Add4toitandthendoubleyouranswer. Nowsubtract6fromthat result and divide your new answer by 2. Write down your answer. Repeat these stepswithanothernumber. Continuewithafewmorenumbers, comparingyournal answerwithyouroriginalnumber. Isthereapatterntoyouranswers?3. Using the four integers 2, 3, 6 and 8 once each in any order and three arithmeticoperationsselectedfromamongaddition, subtraction, multiplication, anddivision, writeexpressionswhosevaluesarethetargetnumbersgivenbelow. Youwillprobablyneedtouseparentheses. Forexample,tohitthetarget90,youcouldwrite90 = (3 + 6) (8 + 2).(a)3 (b)24 (c)36 (d)304. Whendescribingthegrowthof apopulation, thepassageof timeis sometimes de-scribedingenerations, agenerationbeingabout30years. Onegenerationago, youhadtwoancestors(yourparents). Twogenerationsago, youhadfourancestors(yourgrand-parents). Ninety years ago, you had eight ancestors (your great-grandparents). How manyancestorsdidyouhave300yearsago?900yearsago?Doyouranswersmakesense?5. Onarecent episodeof WhoWants toBeaBillionaire, acontestant was askedtoarrangethefollowingvenumbersinincreasingorder. Youtryit,too.(a)2/3 (b)0.6666 (c)3/5 (d)0.666 (e)0.676. Theareaofacirclewhoseradiusisrisgivenbytheexpressionr2. Findtheareaofeachofthefollowingcirclestothenearesttenthofasquareunitofmeasure:(a)acirclewhoseradiusis15cm (b)acirclewhoseradiusis0.3miles7. Choose any number. Double it. Subtract six and add the original number. Now dividebythree. Repeatthisprocesswithothernumbers, until apatterndevelops. Byusingavariablesuchasxinplaceof yournumber, showthatthepatterndoesnotdependonwhichnumberyouchooseinitially.8. Explainwhytherearetwowaystocomputeeachofthefollowing:(a)3(2 + 3 + 5) (b)13(9 + 6 3) (c)(9 + 6 3) 39. Giventheinformationw=4inches andh=7inches, ndtwoways toevaluate2w + 2h. Whatisthegeometricsignicanceofthiscalculation?10. Simplifyx + 2 + x + 2 + x + 2 + x + 2 + x + 2 + x + 2 + x + 2 + x + 2 + x + 2.11. Without resorting to decimals, nd equivalences among the following nine expressions:2 35_35_ 2 3_25_ _25__33_ _53_2 2 532553 1235/2August2012 2 PhillipsExeterAcademyMathematics11. Whatisthevalueof3 + (3)? Whatisthevalueof(10.4) + 10.4? Thesepairsofnumbersarecalledopposites. Whatisthesumofanumberanditsopposite?Doeseverynumberhaveanopposite?Statetheoppositeof:(a) 2.341 (b)1/3 (c)x (d)x + 2 (e)x 22. Asshownonthenumberlinebelow,krepresentsanunknownnumberbetween2and3. Ploteachofthefollowing,extendingthelineifnecessary:(a)k + 3 (b)k 2 (c) k (d)6 k3 2 1 0 1 2 3 4 5 6 k3. Youarealreadyfamiliar with operationsinvolvingpositive numbers,butmuchmath-ematicalworkdealswithnegativenumbers. Commonusesincludetemperatures, money,andgames. Itisimportanttounderstandhowthesenumbersbehaveinarithmeticcalcu-lations. First,consideradditionandsubtraction. Foreachofthefollowing,showhowtheanswercanbevisualizedusinganumber-linediagram:(a)Theairtemperatureat2pmwas12. Whatwastheairtemperatureat8pm, ifithaddropped15bythen?(b) Telescope Peak in the Panamint Mountain Range, which borders Death Valley, is 11045feetabovesealevel. Atitslowestpoint,DeathValleyis282feetbelowsealevel. WhatistheverticaldistancefromthebottomofDeathValleytothetopofTelescopePeak?(c)Inarecentgame, Ihadascoreof3. Ithenproceededtolose5pointsand7pointsonmynexttwoturns. Ontheturnafterthat,however, Igained8points. Whatwasmyscoreatthismomentinthegame?4. Tobuyaticketforaweeklystatelottery, apersonselects6integersfrom1to36,theordernotbeingimportant. Thereare1947792suchcombinationsofsixdigits. Alexandnine friends want towinthe lotterybybuyingeverypossible ticket (all 1947792combinations), andplantospend16hours adaydoingit. Assume that eachpersonbuysoneticketeveryveseconds. Whatdoyouthinkofthisplan? Cantheprojectbecompletedwithinaweek?5. Locatethefollowingnumbersrelativetoeachotheronanumberline:(a)3.03 (b)3.303 (c)3.033 (d)3.333 (e)3.336. Theareaof thesurfaceof asphereisdescribedbytheformulaS=4r2, whereris theradius of thesphere. TheEarthhas aradius of 3960miles anddrylandformsapproximately29.2%oftheEarthssurface. WhatistheareaofthedrylandonEarth?WhatisthesurfaceareaoftheEarthswater?7. Markarandomnumberxbetween1and2(ataspotthatonlyyouwillthinkof)onanumberline. Plottheoppositeofeachofthefollowing:(a)x (b)x + 5 (c)x 4 (d)6 xAugust2012 3 PhillipsExeterAcademyMathematics11. At186282milespersecond,howfardoeslighttravelinayear?Giveyouranswerinmiles, butusescienticnotation, whichexpressesanumberlike93400000as9.34 107(whichmight appear onyour calculator as 9.34E7instead). Ayear is approximately365.25days. Theanswertothisquestioniscalledalight yearbyastronomers, whouseittomeasurehugedistances. OtherthantheSun, thestarnearesttheEarthisProximaCentauri,amere4.2lightyearsaway.2. Before you are able to take a bite of your new chocolate bar,a friend comes along andtakes1/4of thebar. Thenanotherfriendcomesalongandyougivethisperson1/3ofwhatyouhaveleft. Makeadiagramthatshowsthepartofthebarleftforyoutoeat.3. Lateryouhaveanotherchocolatebar. Thistime,afteryougiveaway1/3ofthebar,afriendbreakso3/4oftheremainingpiece. Whatpartoftheoriginalchocolatebardoyouhaveleft?Answerthisquestionbydrawingadiagram.4. ProtsfortheWhirligigSportsEquipmentCompanyforsixscalyears,from1993through1998, are graphedat right. The vertical scaleisinmillionsofdollars. Describethechangeinprotfrom(a)1993to1994;(b) 1994to1995;(c)1997to1998.Duringthesesixyears, didthecompanymakean overall prot or sustain an overall loss?Whatwasthenetchange?................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................24242.61.51.71.80.62.393 94959697 985. The temperature outside is dropping at 3 degrees per hour. Given that the temperatureatnoonwas0,whatwasthetemperatureat1pm?at2pm?at3pm?at6pm?Whatwasthetemperaturethoursafternoon?6. Thisyear, thereare1016studentsattheAcademy, ofwhom63liveinDunbarHall.Tothenearesttenthofapercent,whatpartofthestudentpopulationlivesinDunbar?7. Let krepresent some unknownnumber between 4and 5. Locatebetweentwoconsecutiveintegerseachofthefollowing:(a) k (b) k + 5 (c)k2+ 2 (d)k + 22August2012 4 PhillipsExeterAcademyMathematics11. Use the balancediagram below to nd how many marbles it takes to balance one cube.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .................................................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .................................................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ......2. (Continuation)Usingctostandfortheweightof onecubeandmfortheweightofonemarble,writeanequationthatmodelsthepictureinthepreviousproblem. Usethisequationtondhowmanymarblesittakestobalanceonecube.3. The division problem 1234is equivalent to the multiplication problem 1243 . Explain.Writeeachofthefollowingdivisionproblemsasequivalentmultiplicationproblems:(a)20 5 (b)20 15(c)20 25(d)a bc(e)bc a4. Whatisthevalueof23 32 ? Whatisthevalueof4 14 ? Thesepairsofnumbersarecalled reciprocals. What is the product of a number and its reciprocal?Does every numberhaveareciprocal?Statethereciprocalofthefollowing:(a)53(b) 12(c)2000 (d)ab(e)1.2 (f )x5. Hereisanothernumberpuzzle: Pickanumber, add5andmultiplytheresultby4.Addanother 5andmultiplytheresult by4again. Subtract 100fromyour result anddivideyouranswerby8. Howdoesyouranswercomparetotheoriginal number? Youmayneedto doa coupleof exampleslike this untilyou seethe pattern. Use avariable forthechosennumberandshowhowthepatternholdsforanynumber.6. (Continuation)Makeupanumberpuzzleofyourown. Beabletoverifythepatternusingavariableforthenumberchoseninitially.7. Jesstakesaboardthatis50incheslongandcutsitintotwopieces, oneofwhichis16incheslongerthantheother. Howlongiseachpiece?8. Consider thesequence of numbers 2, 5, 8, 11, 14, . . . , inwhicheachnumber is threemorethanitspredecessor.(a)Findthenextthreenumbersinthesequence.(b) Findthe100thnumberinthesequence.(c)Usingthevariablentorepresentthepositionofanumberinthesequence, writeanexpression that allows you to calculate the nthnumber. The 200thnumber in the sequenceis599. Verifythatyourexpressionworksbyevaluatingitwithnequalto200.9. Agroupoftenpersonswereplanningtocontributeequal amountsofmoneytobuysomepizza. Afterthepizzawasordered, onepersonleft. Eachoftheotherninepersonshadtopay60centsextraasaresult. Howmuchwasthetotalbill?August2012 5 PhillipsExeterAcademyMathematics11. Inthebalancediagrambelow,ndthenumberofmarblesthatbalanceonecube.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .................................................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .................................................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ......2. Foreachofthefollowing,ndthevalueofxthatmakestheequation true. Theusualwayofwordingthisinstructionissolveforx:(a)2x = 12 (b) 3x = 12 (c)ax = b3. On each of the following number lines, all of the labeled points are evenly spaced. Findcoordinatesforthesevenpointsdesignatedbytheletters.3 a b c d 23p 8/3 q 6 r4. Using the four integers 1, 2, 3 and 4 once each in any order and three arithmeticoperationsselectedfromamongaddition, subtraction, multiplication, anddivision, isitpossible towrite anexpression whosevalue is 1?Usingthe samenumbersand conditions,howmanyoftheintegersfrom1to10canyouform?Youwillneedtouseparentheses.5. Arectanglewhoselengthisxandwhosewidthis1iscalledanx-block. Thegureshowstwoofthem.(a)Whatistheareaofanx-block?(b) Whatisthecombinedareaoftwox-blocks?(c)Showthattherearetwodierentwaystocombinetwox-blockstoformarectanglewhoseareais2x.(d) Drawtwodierentrectangulardiagramstoshowthatx + 2x = 3x.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................1x1x6. Usethedistributivepropertytoexplainwhy3x + 2xcanbesimpliedto5x.7. (Continuation)Writeeachofthefollowingasaproductofxandanotherquantity:(a)16x + 7x (b)12x 6x (c)ax + bx (d)px qx8. Solveeachofthefollowingequationsforx:(a)16x + 7x = 46 (b)12x 6x = 3 (c)ax + bx = 10 (d)px qx = r9. Draw a balance diagram that is modeled by the equation c+m+c+7m+c = 2c+2m+3c.Howmanymarbleswillonecubebalance?10. Youhaveseenthatmultiplicationdistributesoveraddition. Doesmultiplicationdis-tributeoversubtraction? Doesmultiplicationdistributeovermultiplication?Doesmulti-plicationdistributeoverdivision?Useexamplestoillustrateyouranswers.August2012 6 PhillipsExeterAcademyMathematics11. In baseball statistics, a players slugging ratio is dened to bes + 2d + 3t + 4hb, wheres is the number of singles, d the number of doubles, t the number of triples and h the numberofhomerunsobtainedinbtimesatbat. Danacametobat75timesduringtheseason,andhit12singles, 4doubles, 2triples, and8homeruns. WhatisDanassluggingratio,roundedtothreedecimalplaces?2. Make a dot somewhere between0 and 0.5 on a number line, and label it k. Place eachofthefollowingonthesamenumberlineasaccuratelyasyoucan.(a) k (b)2k (c)k2(d)k 2 (e)k3. Simplifyeachofthefollowing:(a)thesumof6x + 2and 8x + 5;(b) theresultofsubtracting5x 17from8x + 12;(c)theproductof7xand4x 9.4. Solve23 (3x + 14) = 7x+6, by rst multiplying both sides of the equation by 3, beforeapplyingthedistributiveproperty.5. Because12x2+5x2isequivalentto17x2,theexpressions12x2and5x2arecalled liketerms. Explain. Whyare12x2and5xcalledunliketerms?Are3aband11abliketerms?Explain. Are12x2and5y2liketerms?Explain. Are12x2and12xliketerms?Explain.6. In each of the following, use appropriate algebraic operations to remove the parenthesesandcombineliketerms. Leaveyouranswersinasimpleform.(a)x(2x) + 2(x + 5) (b)2x(5x 2) + 3(6x + 7) (c)5m(3m2n) + 4n(3m2n)7. True or false, with justication:712 + 1112 +112 + 1912is equivalent to112(7+11+1+19).8. JesshasjustnishedtellingLeeaboutlearningawonderfulnewalgebratrick: 3 +5xcanbesimpliedveryneatlytojust8x,becausea + bxisthesameas(a + b)x. NowLeehastobreaksomebadnewstoJess. Whatisit?9. Findwholenumbersmandnthatttheequation3m + 6n=87. Isitpossibletondwholenumbersmandnthatttheequation3m + 6n = 95?Ifso,ndanexample.Ifnot,explainwhynot.10. Ifmandnstandforintegers,then2mand2nstandforevenintegers. Explain. Usethedistributivepropertytoshowthatthesumofanytwoevennumbersiseven.11. (Continuation)Showthatthesumofanytwooddnumbersiseven.12. Solve9x + 2 =34(2x + 11).August2012 7 PhillipsExeterAcademyMathematics11. Thedistributivepropertystatesthat(1)x + 1xisthesameas(1 + 1)x, andthisis0. Itfollowsthat(1)xisthesameas x. Explainwhy,thenusesimilarreasoningtoexplain why (x)yis the same as (xy). By the way, is it correct to say, x is a negativenumber?2. Simplifythe expressionk 2(k (2 k)) 2 bywriting it withoutusingparentheses.3. LastyearthepriceofaniPodwas$240.(a)Thisyearthepriceincreasedto$260. Bywhatpercentdidthepriceincrease?(b) Ifthepricenextyearwere5%morethanthisyearsprice,whatwouldthatpricebe?(c)Ifthepricedropped5%theyearafterthat, showthatthepricewouldnotreturnto$260. Explaintheapparentparadox.4. DuringarecentepisodeofWhoWantstoBeaBillionaire, yourfriendTerrycalledyou,needinghelpwith solving theequation 5x +1 = 2x +7. Write downthe step-by-stepinstructionsyouwouldgiveTerryoverthephone.5. Whichnumberisclosertozero, 4/5or5/4?6. SeveralPrepswere meetingin aroom. After45 ofthem left,the room was 5/8as fullasitwasinitially. HowmanyPrepswereintheroomatthestartofthemeeting?7. The gure shows some more algebra blocks. The 1-by-1 squareis called a unitblock, or a 1-block. Below the 1-block is a represen-tationofx + 2, formedfromanx-blockandtwo1-blocks. Drawadiagramusingtheappropriatenumberofx-blocksand1-blockstoillustratethedistributivepropertry3(x + 2) = 3x + 6..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................x1 1118. Often it is necessary to rearrange an equation so that one variable is expressed in termsof others. Forexample, theequationD=3texpressesDintermsof t. ToexpresstintermsofD,dividebothsidesofthisequationby3toobtainD/3 = t.(a)SolvetheequationC= 2rforrintermsofC.(b) Solvetheequationp = 2w + 2hforwintermsofpandh.(c)Solvetheequation3x 2y= 6foryintermsofx.9. Onanumberline,whatnumberishalfwaybetween(a) 4and11? (b)mandn?10. Coeebeanslose12.5%oftheirweightduringroasting. Inordertoobtain252kgofroastedcoeebeans,howmanykgofunroastedbeansmustbeused?11. Theproductof twonegativenumbersisalwaysapositivenumber. Howwouldyouexplain this rule to a classmate whodoes not understandwhy the productof two negativenumbersmustbepositive?August2012 8 PhillipsExeterAcademyMathematics11. TemperatureismeasuredinbothCelsiusandFahrenheitdegrees. Thesetwosystemsareofcourserelated: theFahrenheittemperatureisobtainedbyadding32to9/5oftheCelsiustemperature. Inthefollowingquestions, letCrepresenttheCelsiustemperatureandFtheFahrenheittemperature.(a)WriteanequationthatexpressesFintermsofC.(b) UsethisequationtondthevalueofFthatcorrespondstoC= 20.(c)OntheCelsiusscale, waterfreezesat0andboilsat100. UseyourformulatondthecorrespondingtemperaturesontheFahrenheitscale. Doyourecognizeyouranswers?(d) A quick way to get an approximate Fahrenheit temperature from a Celsius temperatureis to double the Celsius temperature and add 30. Explain why this is a good approximation.Convert23Celsiusthequickway. Whatisthedierencebetweenyouranswerandthecorrectvalue?ForwhatCelsiustemperaturedoesthequickwaygivethecorrectvalue?2. You measureyour strideandndittobe27 inches. If youweretowalk toNewelds,atown4.5milesnorthof Exeter, howmanystepswouldyouhavetotake? Rememberthatthereare12inchesinafoot,3feetinayard,and5280feetinamile.3. TheMillersmustmakea70-mileThanksgivingtriptovisittheirgrandparents. PatMillerbelievesindrivingatasteadyrateof50milesperhour.(a)WithPatinthedriversseat,howmuchtimewillthetriptake?(b) HowmanymileswilltheMillerstravelin18minutes?(c)Writeanexpressionforthenumberofmilestheywillcoverintminutesofdriving.(d) Aftertminutesofdriving,howmanymilesremaintobecovered?4. The length of a certain rectangle exceeds its width by exactly 8 cm, and the perimeteroftherectangleis66cm. Whatisthewidthoftherectangle?Althoughyoumaybeableto solve this problem using a method of your own, try the following approach, which startsbyguessingthewidthof therectangle. Studytherstrowof thetablebelow, whichisbasedona10-cmguessforthewidth. Thenmakeyourownguessanduseittollinthenextrowofthetable. Ifyouhavenotguessedthecorrectwidth, useanotherrowofthetableandtryagain.guess length perimeter target check?10 10 + 8 = 18 2(10) + 2(18) = 56 66 noNow use the experience gained by lling in the table to write an equation for the problem:Writewintheguesscolumn, ll inthelengthandperimeterentriesintermsofw, andsetyourexpressionfortheperimeterequal tothetargetperimeter. Solvetheresultingequation. Thisapproachtocreatingequationsiscalledtheguess-and-checkmethod.5. Solveforx: (a)3x 4 = 11 (b) 2x + 5 = 1 (c)ax + b = cAugust2012 9 PhillipsExeterAcademyMathematics11. Number-line graphs. Observe the following conventions, which may already be familiar: Toindicateanintervalonthenumberline,thickenthatpartofthenumberline. To indicate that an endpoint of an interval is included, place a solid dot on the number. Toindicatethatanendpointisnotincluded,placeanopencircleonthenumber.Forexample, thediagramillustratesthosenumbersthataregreaterthan 2andlessthanorequalto3. 2 3Drawanumberlineforeachofthefollowingandindicatethenumbersdescribed:(a)Allnumbersthatareexactlytwounitsfrom5.(b) Allnumbersthataremorethantwounitsfrom5.(c)Allnumbersthataregreaterthan 1andlessthanorequalto7.(d) Allnumbersthatarelessthanfourunitsfromzero.2. Percentpractice: (a)25%of200iswhatnumber?(b)200is25%ofwhatnumber?(c)Express2/25asadecimal;asapercent. (d)Express24%asadecimal;asafraction.3. At West Point, theplebe(rst yearcadet) whobringsdessert tothetablemustdivideitintopiecesthatareexactlythesizerequestedbythecadetsatthetable. Onenight, thetwoseniorsassignedtothetablerequested1/6of thepieand1/5of thepie,respectively. Howmuchofthepiedidthatleavefortheyoungercadets?4. Ryanearnsx dollarseverysevendays. Write anexpressionforhowmuchRyanearnsin one day. Ryans spouse Lee is paid twice as much as Ryan. Write an expression for howmuchLeeearnsinoneday. Writeanexpressionfortheircombineddailyearnings.5. Solveforx: (a)2(x 3) = 4 (b) 3(2x + 1) = 5 (c)a(bx + c) = d6. Daystudent Averyjust bought 10gallons of gasoline, theamount of fuel usedforthelast355milesof driving. Beingacurioussort, Averywonderedhowmuchfuel hadbeenusedincitydriving(whichtakesonegallonforevery25miles)andhowmuchhadbeenusedinfreewaydriving(whichtakesonegallonforeach40miles). Averystartedbyguessing6 gallons for the city driving, then completed the rstrow of the guess-and-checktablebelow. Noticethefailedcheck. Makeyourownguessanduseittoll inthenextrowofthetable.cityg freewayg citymi freewaymi total mi target check6 10 6 = 4 6(25) = 150 4(40) = 160 150 + 160 = 310 355 noNowwritec inthecity-galloncolumn, ll intheremainingentries interms of c, andsetyourexpressionforthetotal mileageequal tothetargetmileage. Solvetheresultingequation.7. Onanumberline,graphallnumbersthatarecloserto5thantheyareto8.August2012 10 PhillipsExeterAcademyMathematics11. Remywalkedtoafriendshouse, mmilesaway, atanaveragerateof 4mph. Them-milewalkhomewasatonly3mph,however. Expressasafraction(a)thetimeRemyspentwalkinghome;(b) thetotaltimeRemyspentwalking.2. Thesumoffourconsecutiveintegersis2174. Whataretheintegers?3. (Continuation)Thesmallestoffourconsecutiveintegersisn. Whatexpressionrepre-sents the next larger integer?Write an expression for the sum of four consecutive integers,thesmallestofwhichisn. Writeanequationthatstatesthatthesumoffourconsecutiveintegers is s. Solve the equation for n in terms of s. Check that your answer to the previousquestionsatisesthisequationbyconsideringthecases = 2174.4. Solveforx: (a)2(x 1) = 3(x + 2) (b) 4(2x 2) = 3(x + 1)5. There are three feet in a yard. Find the number of feet in 5 yards. Find the number ofyardsin12feet. Findthenumberoffeetinyyards. Findthenumberofyardsinffeet.6. SamandCamhavealawn-mowingservice. Theirrstjob tomorrow morning is one that usually takes Sam 40 min-utestodoalone,orCam30minutestodoalone. Thistimetheyare goingtoteamup, Samstartingat one sideandCam at the other side. Theproblem is to predicthow manyminutes it will takethemtonishthejob. What part ofthelawnwillSamcompleteinthersttenminutes? Whatpart of the lawn will Cam complete in the rst ten minutes?Whatpartof thelawnwill theteamcompleteintenmin-utes? Setupaguess-and-checktablewithcolumnstitledminutes, Sampart, CampartandTeampart. Whatisthetargetvaluefortheteampart? Fill intworowsofthechartbymakingguessesintheminutescolumn. Thenguessmandcompletethesolutionalgebraically...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................C S7. Writeanexpressionthatrepresentsthenumberthat(a)is7morethanx; (b)is7lessthanx; (c)isxmorethan7;(d)exceedsxby7; (e)isxlessthan7; (f )exceeds7byx.8. Thex2-block,shownat right,is anothermember of the algebra-blockfamily. Draw an algebra-block diagram that shows that x(x+2) = x2+2x.9. There are 396 persons in a theater. If the ratio of women to men is 2:3,and the ratio of men to children is 1:2, how many men are in the theater?.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................xx10. Onanumberline, graphanumberthatistwiceasfarfrom5asitisfrom8. Howmanysuchnumbersarethere?August2012 11 PhillipsExeterAcademyMathematics11. Intervalsonanumberlineareoftendescribedusingthesymbols(greaterthan), (lessthanorequalto),and (greaterthanorequalto). Asyougraphthefollowinginequalities, remembertheendpoint conventionregardingtheuseofthedot andthecircle forincludedandexcludedendpoints,respectively:(a)x < 5 (b)x 6 (c) 12 x (d)4 < x < 8 (e)x < 3or7 x2. SolvetheequationA = P+ Prtforr. SolvetheequationA = P+ PrtforP.3. Usinganumberline, describethelocationofx + y2inrelationtothelocationsofxandy. Isyouransweraectedbyknowingwhetherxandyarepositiveornot?4. Findthe smallestpositive integer divisible byeverypositive integer lessthan or equalto10.5. Evaluate the formula36y+12f+i wheny =2.5, f =2, andi =5. Findaninterpretationforthisformula.6. Theindicatoronthe oiltankinmyhomeindicatedthatthetankwasone-eighthfull.Afteratruckdelivered240gallons of oil,theindicator showedthatthetankwashalf full.Whatisthecapacityoftheoiltank,ingallons?7. Oneof thePEAinterscholasticteamshasstarteditsseasonbadly, winning1game,losing6,andtyingnone. Theteamwillplayatotalof25gamesthisseason.(a)Whatpercentageofthesevengamesplayedsofarhavebeenwins?(b) Starting with its current record of 1 win and 6 losses, what will the cumulative winningpercentagebeiftheteamwinsthenext4gamesinarow?(c)Startingwithits currentrecordof1win and6losses,howmanygamesinarow musttheteamwininorderforitscumulativewinningpercentagetoreachatleast60%?(d) Supposethattheteamwinstenofitsremaining18games. Whatisitsnalwinningpercentage?(e)Howmanyof theremaining18games does theteamneedtowinsothat its nalwinningpercentageisatleast60%? Isitpossiblefortheteamtohaveanal winningpercentageof80%?Explain.8. Graphonanumberlinetheintervalsdescribedbelow:(a)Allnumbersthataregreaterthan1orlessthan 3.(b) Allnumbersthataregreaterthan 5andlessthanorequalto4.(c)Allnumberswhosesquaresaregreaterthanorequalto1.9. Usemathematicalnotationtorepresenttheintervalsdescribedbelow.(a)Allnumbersthataregreaterthan1orlessthan 3.(b) Allnumbersthataregreaterthan 5andlessthanorequalto4.(c)Allnumberswhosesquaresaregreaterthanorequalto1.August2012 12 PhillipsExeterAcademyMathematics11. RandyandSandyhaveatotalof20booksbetweenthem. AfterSandylosesthreebyleavingthemonthebus, andsomebirthdaygiftsdoubleRandyscollection, theirtotalincreasesto30books. Howmanybooksdideachhavebeforethesechanges?2. Combinethefollowingfractionsintoasinglefraction. Expresseachofyouranswersinlowestterms.(a)275+3y4(b)4m523(c)2 +x3(d)x2+2x33x43. Solvethefollowingforx:(a)4 (x + 3) = 8 5(2x 3) (b)x 2(3 x) = 2x + 3(1 x)4. Guessingbirthdays. Pat is working a number trick on Kim, whose birthday is the 29thof February. The table below shows the sequenceof questions that Pat asks, as well as thecalculationsthatKimmakesinresponse. Anothercolumnisprovidedforthealgebrayouaregoingtodotosolvethetrick. Usethelettersmanddformonthandday.Instruction Kim AlgebraWritethenumberofyourbirthmonth 2 mMultiplyby5 10Add7 17Multiplyby4 68Add13 81Multiplyby5 405Addthedayofthemonthofyourbirthday 434After hearing the result of the last calculation, Pat can do a simple mental calculation andthenstateKimsbirthday. Explainhow. Totestyourunderstandingofthistrick, tryitonsomeonewhosebirthdayisunknowntoyou.5. Lastyear,threefthsof theOutingClubweregirls, butthis year thenumberof boysdoubledandsixnewgirlsjoined. Therearenowasmanyboysintheclubastherearegirls. Howmanymembersdidtheclubhavelastyear?6. I amthinkingof nconsecutivepositiveintegers, thesmallestof whichism. Whatformularepresentsthelargestoftheseintegers?7. Place a common mathematical symbol between the numerals 2 and 3, so as to produceanumberthatliesbetween2and3onanumberline.August2012 13 PhillipsExeterAcademyMathematics16 : 206 : 306 : 406 : 507 : 007 : 107 : 201 5 10 15 20 25 30ThegraphdisplaysthetimeofsunsetatExeterduringSeptember. Somequestions:1. Atwhattimedidthesunsetonthe5thofSeptember?onthe30thofSeptember?2. Onwhatdaydoesthesunsetat6:54?at7:08?at6:30?3. Guessthetimeofsunsetonthe1stofOctoberandonthe31stofAugust.4. WhatistheaveragedailychangeofsunsettimeduringthemonthofSeptember?5. Thedotsinthegraphformapattern. JessthinksthatthispatterncontinuesintoOctober, November, andDecember. Whatdoyouthink? Makeagraphthatshowshowthe time of sunsetat Exeter changesduringan entireyear. Agood source for suchdata istheU.S.NavalObservatorysitehttp://aa.usno.navy.mil.6. What happens ontheAutumnal Equinox, whichisthe22ndof September? Guesswhattimethesunrisesonthisday.August2012 14 PhillipsExeterAcademyMathematics11. Aat, rectangular boardis built bygluingtogether anumber ofsquare pieces of the same size. The boardis msquares wide andnsquareslong. Usingthelettersmandn,writeexpressionsfor(a)thetotalnumberof1 1squares;(b) thetotal number of 1 1squareswithfreeedges(thenumber of1 1squaresthatarenotcompletelysurroundedbyothersquares);(c)thenumberofcompletelysurrounded1 1squares;(d) theperimeterofthegure.2. Graph on a number line the intervals corresponding to these two signs on the highway.(a)Themaximumspeedis65mphandtheminimumspeedis45mph.(b) Themaximumspeedis55mph.3. Label thegureatrightsothatitprovidesageometricrepre-sentationofx(x + 3). Noticethatthisquestionisaboutarea.4. Itis sometimes necessaryto write fractions with variables in thedenominator. Without usingyour calculator, rewriteeachof the followingas asinglefraction. Thisiscalledcombiningoveracommondenominator.(a)3a+7a(b)3a+72a(c)3a+7b(d)3 +7b5. Ittakesoneminutetollafour-galloncontainerattheExeterspring. Howlongdoesittaketoll asix-gallon container?Fill in the missing entries in thetablebelow,andplotpointsonthegridatright.1 2 3 4 54 5 6 11 14 19timevolumeNoticethat it makes sensetoconnect thedots youplotted(therebyformingacontinuous pattern). Isthesametrueofthesunset-timegraphyoulookedatrecently?Explain.5101520vol1 2 3 4 time6. Ryan took 25 minutes to type the nal draft of a 1200-word English paper. How muchtimeshouldRyanexpecttospendtypingthenaldraftofa4000-wordHistorypaper?7. Whichofthefollowingsevenexpressionsdoesnotbelonginthelist?a b + c c b + a c (b a) b + a + c a (b c) b (c a) a + c b8. Lastweek,ChrisboughtaDVDfor$10.80whilethestorewashavinga25%-osale.Thesaleisnowover. HowmuchwouldthesameDVDcosttoday?9. Forrest is texting while driving along the freeway at 70 miles per hour. How many feetdoesthecartravelduringthe3-secondintervalwhenForrestseyesarenotontheroad?August2012 15 PhillipsExeterAcademyMathematics11. Thestatementxisbetween13and23denesaninterval usingtwosimultaneousinequalities: 13 < x and x < 23. The statement x is not between 13 and 23 also uses twoinequalities,buttheyarenon-simultaneous: x 13or23 x. Graphthesetwoexamplesonanumberline. Noticethatthereisacompactform13 < x < 23foronlyoneofthem.2. CrossingalongstretchoftheCanadianplains,passenger trains maintain a steady speed of 80 mph.Atthatspeed, whatdistanceiscoveredinhalf anhour?How much time is neededto cover 200 miles?Fill inthemissingentries inthetablebelow, andplotpointsonthegridatright.0 1/2 1 2 3 4 t60 200 300timedistance3. TheproblemsabouttheExeterspringandtheCanadian plains contain relationships that are calleddirect variations. Inyour ownwords, describewhat it means for onequantitytovarydirectlywithanother. Whichofthefollowingdescribedirectvariations?(a)Thegallonsofwaterinatubandthenumberofminutessincethetapwasopened.(b) Theheightofaballandthenumberofsecondssinceitwasthrown.(c)Thelengthofasideofasquareandtheperimeterofthesquare.(d) Thelengthofasideofasquareandtheareaofthesquare.100200300400dist1 2 3 4 time4. (Continuation)Sketchgraphsforeachof thesituationsdescribedabove. Besuretoincludemeaningfuldescriptionsandscalesforeachaxis.5. Remywalkedtoafriendshouse, mmilesaway, atanaveragerateof 4mph. Them-milewalkhomewasonlyat3mph. Remyspent2hourswalkinginall. Findthevalueofm.6. Thesidesofarectangleinthecoordinateplaneareparallel totheaxes. Twoofthevertices of therectangleare(3, 2) and(4, 7). Findcoordinates for theother twovertices. Findtheareaoftherectangle.7. Therectangleshownatrighthasbeenbrokenintofoursmallerrectangles. Theareasofthreeofthe smaller rectangles are showninthe diagram.Findtheareaofthefourthone.8. Ticketstoaschoolplaycost$1.50ifboughtinadvance, and $2.00 if bought at the door. By sellingall 200 of their tickets,the players broughtin $360.Howmanyoftheticketsweresoldinadvance?234270312August2012 16 PhillipsExeterAcademyMathematics11. Chandler was given $75 for a birthday present. This present,along with earnings fromasummerjob, isbeingsetasideforamountainbike. Thejobpays$6perhour, andthebike costs $345. To be able to buy the bike, how many hours does Chandler need to work?2. (Continuation)LethbethenumberofhoursthatChandlerworks. Whatquantityisrepresented by the expression 6h?What quantity is represented by the expression 6h+75?(a)Graphthesolutionstotheinequality6h + 75 345onanumberline.(b) Graphthesolutionstotheinequality6h + 75 < 345onanumberline.Whatdothesolutionstotheinequality6h + 75 345signify?3. Sandyrecentlymadea210-milecartrip, start-ingfromhomeatnoon. Thegraphatrightshowshow Sandys distance from home (measured in miles)dependsonthenumberof hoursafternoon. Makeup a story that accounts for the four distinct parts ofthe graph. In particular, identify the speed at whichSandyspentmostoftheafternoondriving.4. Chasebegananumber puzzlewiththewordsPickanumber, add7toit, anddouble the re-sult.Chasemeanttosay, Pickanumber, doubleit, andadd7totheresult.Arethesetwoinstruc-tionsequivalent?Explain................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ 50100150200dist1 2 3 4 time5. ThedistancefromPEAtothebeachatLittleBoarsHeadis10miles. If youbikefrom PEA tothe beach in 40minutes,what is your averagespeed for the trip?Whatdoesthismean?6. (Continuation)Onthereturntripfromthebeach, youpedal hardfortherst tenminutesand cover 4miles. Tired,you slow down and cover thelast 6 miles in 36 minutes.Whatisyouraveragespeedforthereturntrip?7. Solvetheinequality3 x > 5usingonlytheoperationsofadditionandsubtraction.Isx = 0asolutiontotheinequality?8. Aldenpaidtohavesomeprogramsprintedforthefootball gamelastweekend. Theprintingcostperprogramwas54cents, andtheplanwastosell themfor75centseach.Poorweatherkeptmanyfansawayfromthegame, however, sounluckyAldenwasleftwith100unsoldcopies,andlost$12ontheventure. HowmanyprogramsdidAldenhaveprinted?9. TheMountMajorhikestartsinAltonBay, 716feetabovesealevel. Thesummitis1796feetabovesealevel, andittakesabout45minutesforatypical hikertomaketheclimb. Findtherateatwhichthishikergainsaltitude,infeetperminute.August2012 17 PhillipsExeterAcademyMathematics11. Todoacollegevisit, Wes must makea240-miletripbycar. ThetimerequiredtocompletethetripdependsonthespeedatwhichWesdrives,ofcourse,asthetablebelow shows. Fill in the missing entries, andplot points onthe gridprovided. Dothequantitiestimeandspeedvarydirectly? Itmakessensetoconnectyourplottedpointswithacontinuousgraph. Explainwhy.15 20 25 48 60 r12 8 6 4.8 3speedtime2. PatboughtseveralpensatWalgreens,for60centseach. SpendingthesameamountofmoneyattheBookstore,Patthenboughtafewmorepensthatcost80centseach. Inall,42penswerebought. HowmanypensdidPatbuyatthebookstore?5101520time20 40 60 speed3. Exeterbuildingcodedoesnotpermitbuildingahousethatismorethan35feettall.Anarchitectworkingonthedesignshownatrightwouldliketherooftobeslopedsothatitrises10inchesforeachfootofhorizontalrun.(a)Giventheotherdimensionsinthediagram, will thebuilderbeallowedtocarryoutthisplan?(b) Twoverticalsupports(showndottedinthediagram)aretobeplaced6feetfromthecenterofthebuilding. Howlongshouldtheybe?4. The line through(1, 6) and(0, 3) passes througheveryquadrantexceptone. Whichone?..........................................................................................................................................................................................................................................................................................................................................30225. Combineoveracommondenominatorwithoutusingacalculator:(a)14+15(b)110+111(c)1x+1x + 1Evaluateyouranswer to(c)withx=4andthenwithx=10. Howdotheseanswerscomparetoyouranswersto(a)and(b)?6. Asmall pool is20feetlong, 12feetwideand4feetdeep. Thereare7.5gallonsofwaterineverycubicfoot. Attherateof5gallonsperminute,howlongwillittaketollthispool?7. Shownat right, the y-blockandxy-blockare twomoremembers of thealgebra-blockfamily. Drawanalgebra-blockdiagramthatillustratestheequation(1 + x)y= y + xy.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................xy1yAugust2012 18 PhillipsExeterAcademyMathematics11. The rectangle ABCDshown at righthassidesthat are parallel to thecoordinate axes.Itisthreetimesaswideasitistall, anditsperimeteris56units.(a)FindthelengthandwidthofABCD.(b) Giventhe informationD=(9, 2), ndthecoordinatesforpointsA,B,andC............. . . . . . . . . . .......................AB CD2. Aladderisleaningagainstthesideof abuilding. EachtimeI stepfromonerungtothenext, myfootmoves6inchesclosertothebuildingand8inchesfurtherfromtheground. Thebaseoftheladderis9ftfromthewall. Howfarupthewalldoestheladderreach?3. Eachstepof thestairsleadingfromroom9toroom107intheAcademyBuildinghasaverticalriseof7inchesandahorizontalrunof12inches. Eachstepofthemarblestaircase leading to the Assembly Hall has a vertical rise of 5.5 inches and a horizontal runof13inches.(a)Whichightofstairsdoyouthinkissteeper?Why?(b) Calculatetheratiorise/runforeachightofstairs,andverifythatthegreaterratiobelongstotheightyouthoughttobesteeper.4. (Continuation) The slope of a line is a measure of how steep the line is. It is calculatedbydividingthe change iny-coordinates bythe corresponding change inx-coordinatesbetweentwopointsontheline: slope=changeinychangeinx . Calculatetheslopeofthelinethatgoesthroughthetwopoints(1, 3)and(7, 6). Calculatetheslopeof thelinethatgoesthroughthetwopoints(0, 0)and(9, 6). Whichlineissteeper?5. Explainwhythedescriptionsright5up2,right10up4,left5down2,right5/2up1,andleft1down2/5alldescribethesameinclinationforastraightline.6. Atnoononeday,theExeterRiverpeakedat11feetaboveoodstage. Itthenbegantorecede,itsdepthdroppingat4inchesperhour.(a)At3:30thatafternoon,howmanyinchesaboveoodstagewastheriver?(b) Lettstandforthenumber of hourssincenoon, andhstandforthecorrespondingnumberofinchesthattheriverwasaboveoodstage. Makeatableofvalues, andwriteanequationthatexpresseshintermsoft.(c)Plothversust,puttingtonthehorizontalaxis.(d) Forhowmanyhourspastnoonwastheriveratleast36inchesaboveoodstage?7. Solvethefollowingforx: (a)x2+x5= 6 (b)x3+x + 16= 48. A sign placed at the top of a hill on Route 89 says 8% grade. Trucks use lower gear.Whatdoyouthinkthat8%grademightmean?August2012 19 PhillipsExeterAcademyMathematics11. Jess and Taylor go into the cookie-makingbusiness. Thechartshowshowmanydozensof cookieswerebakedandsold(at$3.50perdozen)duringtherstsixdaysofbusiness.(a)Whatwas their total income duringthosesixdays?(b) Whichwas more protable,the rst threedaysorthelastthreedays?(c)WhatwasthepercentagedecreaseinsalesfromTuesdaytoWednesday? WhatwasthepercentageincreaseinsalesfromWednesdaytoThursday?(d) Thursdayssaleswerewhatpercentofthetotalsales?(e)Onaverage, how many dozensof cookies did Jess and Taylor bake and sell eachday?.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................MonTueWedThuFriSat1224101620142. Theperimeterof arectangleis100anditslengthisx. Whatexpressionrepresentsthewidthoftherectangle?3. Whenathirdof anumber issubtractedfromahalf of thesamenumber, 60istheresult. Findthenumber.4. Suppose that nrepresents aninteger. What expressionrepresents the next largerinteger?thepreviousinteger?thesumofthesethreeconsecutiveintegers?5. Eugeneand Wes are solving the inequality 132 4x 36. Each begins bysubtracting132 from both sides to get 4x 96, and then each divides both sides by 4. Eugene getsx 24 and Wes gets x 24, however. Always happy to oer advice, Alex now suggests toEugeneandWesthatanswerstoinequalitiescanoftenbecheckedbysubstitutingx=0into both the original inequality and the answer. What do you think of this advice?Grapheachoftheseanswersonanumberline. HowdotheresultsofthisquestionrelatetotheoodingoftheExeterRiver?6. (Continuation) After hearing Alexs suggestion about using a test value to check an in-equality, Cameron suggests that the problem could have been done by solving the equation132 4x = 36rst. Completethereasoningbehindthisstrategy.7. (Continuation) Deniz, whohas beenkeepingquiet duringthe discussion, remarks,The only really tricky thing about inequalities is when you try to multiply them or dividethembynegative numbers,butthiskindof stepcanbeavoided altogether. Cameronjusttold us onewayto avoid it,and thereis anotherway, too. Explain thisremark byDeniz.8. Draw the segment from (3, 1) to (5, 6), and the segment from (0, 5) to (2, 0). Calculatetheirslopes. Youshouldnoticethatthesegmentsareequallysteep,andyettheydierinasignicantway. Doyourslopecalculationsreectthisdierence?9. Solvethefollowinginequalityforx: 2(1 3x) (x 5) > 1August2012 20 PhillipsExeterAcademyMathematics11. Eachbeatofyourheartpumpsapproximately0.006literofblood.(a)Ifyourheartbeats50times,howmuchbloodispumped?(b) Howmanybeatsdoesittakeforyourhearttopump0.45liters?2. (Continuation)Direct-variationequationscanbewrittenintheformy=kx, anditiscustomarytosaythatydependsonx. FindanequationthatshowshowthevolumeV pumpeddependsonthenumberofbeatsn. Graphthisequation,usinganappropriatescale,andcalculateitsslope. Whatdoesthesloperepresentinthiscontext?3. Estimatetheslopesofallthesegmentsinthedia-gram. Identifythosewhoseslopesarenegative. Findwordstocharacterizelinesthathavenegativeslopes.4. Find the slope of the line containing the points(4, 7)and(6, 11). Findcoordinatesfor another pointthatliesonthesamelineandbepreparedtodiscussthemethodyouusedtondthem.5. Findaneasywaytodothefollowingcalculationsmentally: (a)25 39 4 (b)63250.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .abcd136. ToearnHallofFamedistinctionatPEA,agirl onthecross-countryteammustrunthe 5-km course in less than 20 minutes. What is the average speed of a 20-minute runner,inkmperhour?inmeterspersecond?Expressyouranswerstotwodecimalplaces.7. (Continuation)Theproportion520=x60ishelpfulforthepreviousquestion. Explainthisproportion,andassignunitstoallfourofitsmembers.8. The diagram shows the last member of the algebra-block family,the y2-block. Show how an xy-block and a y2-block can be combinedtoillustratetheequation(x + y)y = xy + y2.9. Whichisgreater,73percentof87,or87percentof73?..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................yy10. Coreydeposits$300inabankthatpays4%annualinterest. HowmuchinterestdoesCoreyearninoneyear?Whatwouldtheinterestbeiftheratewere6%?11. Alexwashiredtounpackandclean576verysmall itemsof glassware, atvecentsperpiecesuccessfullyunpacked. Foreveryitembrokenduringtheprocess,however,Alexhadtopay$1.98. Attheendofthejob,Alexreceived$22.71. HowmanyitemsdidAlexbreak?August2012 21 PhillipsExeterAcademyMathematics11. Eachofthedatasetsatrightrepresentspointsonaline.Inwhichtableisonevariabledirectlyrelatedtotheother?Whydoes the other table not represent adirect variation?Fillinthemissingentryineachtable.2. (Continuation)Plotthedatafromthetablesinthepre-viousquestiononthesamesetof axesandusearulertodrawalinethrougheachsetofpoints. Bylookingatthegraph, howcouldyourecognizethedirectvariation? Whatsimilaritiesanddierencesaretherebetweenthetwolinesdrawn?161040x19104y161040x1560y3. Supposethat nrepresents apositiveeveninteger. What expressionrepresents thenext eveninteger?the nextodd integer?I am thinking of three consecutive evenintegers,whosesumis204. Whatarethey?4. Acarandasmall truckstartedoutfromExeterat8:00am. Their distances from Exeter, recorded at hourly intervals,arerecordedinthetablesatright. Plotthisinformationonthe samesetof axes and drawtwo linesconnectingthepointsineachsetofdata. Whatistheslopeofeachline? Whatisthemeaningoftheseslopesinthecontextofthisproblem? 12 : 0011 : 0010 : 009 : 008 : 00time208156104520car18413892460truck5. (Continuation) Let t be the number of hours each vehicle has been traveling since 8:00am(thust=0means8:00am), andletdbethenumberofmilestraveledafterthours.Foreachvehicle,writeanequationrelatingdandt.6. Day studentChris does a lot of babysitting. Whenparents drop o their children andChriscansuperviseathome, thehourlyrateis$3. If Chrishastotravel tothechildshome,thereisaxedchargeof$5fortransportationinadditiontothe$3hourlyrate.(a)Graphy=3xandy=3x + 5. Whatdotheselineshavetodowiththebabysittingcontext?Whatfeaturedotheyhaveincommon?Howdotheydier?(b) Whatdoes the graph of y= 3x +6look like?Whatchange in the babysitting contextdoesthislinesuggest?7. If kstandsforaninteger, thenisitpossiblefork2+ ktostandforanoddinteger?Explain.8. Canyouthinkofanumberkforwhichk2< kistrue?Graphallsuchnumbersonanumberline. Alsodescribethemusingwords,andusingalgebraicnotation.9. OneyearafterRobindeposits400dollarsinasavingsaccountthatpaysr%annualinterest,howmuchmoneyisintheaccount?Writeanexpressionusingthevariabler.10. Solvex4+x + 1312andshadethesolutionintervalonanumberline.11. Findthreeconsecutiveoddnumberswhosesumis117. Findtwowaystodothis.August2012 22 PhillipsExeterAcademyMathematics11. If you graph the line y= 0.5x+3 on your calculator, it is likely that both axis interceptsarevisible. Ifyoutrytography= 0.1x +18onyourcalculator,itisquitelikelythattheaxis intercepts are not both visible. What are the axis intercepts?Describe how to set thecalculatorswindowsothattheybothbecomevisible.2. How muchtime does ittake for ajettogo119 miles,if its speedis 420mph?Be suretospecifytheunitsforyouranswer.3. Wordchains. Astheancientalchemistshoped, itispossibletoturnleadintogold.You change one letter at a time, always spelling real words: leadloadtoadtoldgold.Usingthesametechnique,showhowtoturnworkintoplay.4. Findcoordinatesforthepointswheretheline3x + 2y= 12intersectsthex-axisandthey-axis. Thesepointsarecalledthex-interceptandy-intercept,respectively. Usethesepointstomakeaquicksketchoftheline.5. Drivers in distress near Exeter have two towing services to choose from: Brooks BodyShop charges$3 per mile for the towing,and a xed $25 chargeregardless of the length ofthe tow. Morgan Motors charges a at $5 per mile. On the same system of axes, representeachofthesechoicesbyalineargraphthatplotsthecostofthetowversusthelengthofthetow. Ifyouneededtobetowed,whichservicewouldyoucall,andwhy?6. Comparethegraphofy= 2x + 5withthegraphofy= 3x + 5.(a)Describeacontextfromwhichtheequationsmightemerge.(b) Linear equations that looklikey =mx + baresaidtobeinslope-intercept form.Explain. Theterminologyreferstowhichofthetwointercepts?7. DrivingfromBostontoNewYorkoneday,Sashacoveredthe250milesinvehours.Because of heavy trac, the 250-mile return took six hours and fteenminutes. Calculateaveragespeeds forthetriptoNewYork, thetripfromNewYork, andtheroundtrip.Explainwhytheterminologyaveragespeedisabitmisleading.8. Findthevalueofxthatmakes0.1x + 0.25(102 x) = 17.10true.9. Sothatitwillbehandyforpayingtollsandparkingmeters, Leeputspocketchange(dimesandquartersonly)intoacupattachedtothedashboard. Therearecurrently102coinsinthecup,andtheirmonetaryvalueis$17.10. Howmanyofthecoinsaredimes?10. Findallthevaluesofxthatmake0.1x + 0.25(102 x) < 17.10true.11. Withoutusing parentheses,write an expression equivalent to 3(4(3x6) 2(2x+1)).12. Oneyear after RobindepositsPdollarsinasavingsaccount that paysr%annualinterest,how muchmoney is in the account?Write an expression in terms of the variablesPandr. Ifyoucan,writeyouranswerusingjustasingleP.August2012 23 PhillipsExeterAcademyMathematics11. DaystudentMorganlefthomeat7:00onemorning,determinedtomaketheten-miletriptoPEAonbicycleforachange. Soonthereafter, aparentnoticedforgottenmathhomeworkonthekitchentable, gotintothefamilycar, andtriedtocatchupwiththeforgetful child. Morgan had a fteen-minute head start, and was pedaling at 12 mph, whiletheparentpursuedat30mph. WasMorganreunitedwiththehomeworkbeforereachingPEAthatday?Ifso,where?Ifnot,atwhattimeduringrstperiod(math,whichstartsat8:00)wasthehomeworkdelivered?2. Farmer MacGregor needstoputa fencearound arectangular carrot patch thatis oneandahalftimesaslongasitiswide. Theprojectuses110feetoffencing. Howwideisthegarden?3. Combineoveracommondenominator:1a+23a+ 34. Conrmthatthevepointsinthetablealllieonasingleline.Write an equation for the line. Use your calculator to make a scatterplot,andgraphthelineonthesamesystemofaxes.5. If6%ofxisthesameas5%of120,thenwhatisx? 10123x11357y6. Findandgraphthesolutionsetsfortherelatedquestions:(a)46 3(x + 10) = 5x + 20(b)46 3(x + 10) < 5x + 20 (c)46 3(x + 10) > 5x + 207. At1pm, youstartoutonyourbikeat12mphtomeetafriendwholives8milesaway. Atthesametime,thefriendstartswalkingtowardyouat4mph. Howmuchtimewillelapsebeforeyoumeetyourfriend?Howfarwillyourfriendhavetowalk?8. The population of a small town increased by 25% two years ago and then decreased byby25%lastyear. Thepopulationisnow4500persons. Whatwasthepopulationbeforethetwochanges?9. Giventhat it costs $2.75less tobuyadozendoughnuts thantobuytwelvesingledoughnuts,andthat65doughnutscost$25.25,andthat65 = 5 12 + 5,whatisthepriceofasingledoughnut?10. ThevolumeofacircularcylinderisgivenbytheformulaV= r2h.(a)Findthevolumeofacylinderthathasa15-cmradiusandis12cmhigh.(b) Solve thevolume formulaforh,and thenndh,given thatthe volume is 1000cc andtheradiusis10cm.11. It takes ten Preps ten days to paint ten houses. How many houses can ve Preps paintinvedays?August2012 24 PhillipsExeterAcademyMathematics11. Whichofthefollowingpairsofquantitiesvarydirectly?(a)thecircumferenceofacircleandthediameterofthecircle;(b) thedistancetraveledintwohoursandthe(average)rateoftravel;(c)thenumberofgallonsofgasolineboughtandthecostofthepurchase;(d) theareaofacircleandtheradiusofthecircle.2. Ajet,cruisingat26400feet,beginsitsdescentintoLoganAirport,whichis96milesaway. Another jet, cruisingat 31680feet, is 120miles fromLoganwhenit begins itsdescent. Whichofthesetwopathsofdescentissteeper?Explain.3. The diagram shows two steel rods hinged at one end. The other end is connected by abungee cord (the dotted segment),whose unstretchedlengthis 10inches. The rods are 5inches and18inches long. Use inequalitysymbols todescribe allthepossiblelengthsforthebungeecord,whichstaysattachedatbothendswhileitisbeingstretched.......................................................................................................................................................................................................................................................................................................................................................................................................................................... . . . .. . . . . .. . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .1854. AccordingtotheUS CensusDepartment,someonebornin 1950hasalifeexpectancyof68.2years,whilesomeonebornin1970hasalifeexpectancyof70.8years.(a)Whatisareasonablelifeexpectancyofsomeonebornin1960?(b) Whatisareasonablelifeexpectancyofsomeonebornin1980?(c)Whatisareasonablelifeexpectancyofsomeonebornin2000?Part (a) is an interpolation question. Parts (b) and (c) are extrapolationquestions. Whichofyouranswersareyouthemostcondentabout?Explain.5. Multiply2 + xby2x. Drawanalgebra-blockdiagramtoillustratethiscalculation.6. Whenitis150mileswestofitsdestination, ajetisyingat36920feet. Whenitis90miles westof its destination,thejetis at21320 feet. Usingthis data,sketcha graphofthejetsdescent. Isalinearmodelreasonabletouseinthissituation?Explain.7. For each of the following situations, draw a plausible graph that shows the relationshipbetween the time elapsed (horizontal axis) and the indicated speed (vertical axis). In otherwords,graphspeedversustimeforeachofthefollowing:(a)Acarinabumpertesttravelsatasteadyspeeduntilitcrashesintoawall.(b) Yourworkoutconsistsofsomejogging,somehardrunning,somemorejogging,somemorehardrunning,andnallysomewalking.(c)Arollercoasterslowlyclimbsupasteeprampandthenzoomsdowntheotherside.(Plotthecarsspeedjusttothebottomofthersthill.)(d) Acarspeedsatasteadyratealongahighwayuntil anocerpullsitoverandgivesthedriveraticket. Thecarthenresumesitsjourneyatamoreresponsiblespeed.8. Solvethefollowinginequalitiesandshadetheirsolutionintervalsonanumberline.(a)2x3+3x + 52 5 (b)12(x 1) + 3 >13(2x + 1) 1August2012 25 PhillipsExeterAcademyMathematics11. Asquaregameboardisdividedintosmallersquares,whicharecoloredredandblackas on a checkerboard. All four corner squares are black. Let r and b stand for the numbersofredandblacksquares,respectively. Whatisthevalueoftheexpressionb r?2. Atnoon,myodometerread6852miles. At3:30pm,itread7034miles.(a)Whatwasmyaveragerateofchangeduringthesethreeandahalfhours?(b) Lettrepresentthenumberof hoursIhavebeendrivingsincenoonandyrepresentmyodometerreading. Writeanequationthatrelatesyandt. Assumeconstantspeed.(c)Graphyourequation.(d) Showthat the point (5,7112) is onyour line, andtheninterpret this point inthecontextofthisproblem.3. Whatistheslopebetween(3, 7)and(5, 4)? (5, 4)and(3, 7)? (a, b)and(c, d)? (c, d)and(a, b)?4. Ontopofaxedmonthlycharge,Averys cellphonecompanyaddsafeeforeachtextmessagesent. AverysJunebillwas$50.79,whichcovered104textmessages. ThebillforMay,whichcovered83textmessages,wasonly$46.59.(a)Whatisthepriceofatextmessage?(b) Whatisthexedmonthlycharge?(c)WhatwouldAverybechargedforamonththatincluded200textmessages?(d) WhatwouldAverybechargedforamonththatincludedntextmessages?5. AfriendsuggestedthatI changemycellphonecompany. Thisnewcompanyhasaxedmonthlychargeof$39.99,butitchargesonly12centsforeachtextmessage. Isthisabetterdealthantheonedescribedinthepreviousproblem?Giveevidence.6. Forwhatvaluesofxwill thesquareandtherect-angleshownatrighthavethesameperimeter?7. Thepoint(3, 2)isontheliney= 2x +b. Findthevalueofb. Graphtheline.8. Are (2, 9) and (3, 6) both on the line y= 4x+6?Ifnot,ndanequationforthelinethatdoespassthroughbothpoints.x + 5 x + 3x + 79. Afteryougraphtheliney= 4x + 6,nd(a)they-coordinateofthepointonthelinewhosex-coordinateis2;(b) thex-coordinateofthepointonthelinewhosey-coordinateis2.10. In each of the following, describe the rate of change between the rst pair and the sec-ond,assumingthattherstcoordinate ismeasuredinminutesandthesecondcoordinateismeasuredinfeet. Whataretheunitsofyouranswer?(a)(2, 8)and(5, 17) (b)(3.4, 6.8)and(7.2, 8.7) (c)_32 , 34_and_14 , 2_August2012 26 PhillipsExeterAcademyMathematics11. If you double all the sides of a square, a larger square results. By what percentage hastheperimeterincreased?Bywhatpercentagehastheareaincreased?2. Giventhe venumbers8/25,13/40, 19/60,33/100, and59/180, ndthetwothatareclosesttogetheronanumberline,andndthedistancebetweenthem.3. Findthex-interceptandthey-interceptoftheequationy= 32x + 6. Graph.4. Thegraphshowshowthelength(measuredincm) of a pendulum is related to the time (measuredinsec)neededforthependulumtomakeonecom-pleteback-and-forthmovement(whichiscalledtheperiod). Find the length of a pendulum that swingstwiceasoftenasa30-cmpendulum.5. Howfarapartonanumberlineare(a)12and18?(b)12and 7?(c) 11and 4?6. Atoymanufacturerisgoingtoproduceanewtoycar. Eachonecosts$3tomake, andthecom-panywillalsohavetospend$200tosetupthemachinerytomakethem.(a)Whatwillitcosttoproducethersthundredcars?therstncars?(b) Thecompanysellsthecarsfor$4each. Thusthecompanytakesin$400bysellingonehundredcars. Howmuchmoneydoesthecompanytakeinbysellingncars?(c)Howmanycarsdoesthecompanyneedtomakeandsellinordertomakeaprot?..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................10 20 30len0.51.0time7. Whatisthedistancebetween6and 6? between24and17? between17and24?betweent and4? Thedistancebetweentwopoints is always positive. If aandb aretwopointsonanumberline, thedistanceisthereforeeithera borb a, whicheverisnonnegative. This is an example of an absolute-valuecalculation, and the resultis written|a b|. Whatisthemeaningof |b a|?8. Acyclistrides30kmatanaveragespeedof9km/hr. Atwhatratemustthecyclistcoverthenext10kminordertobringtheoverallaveragespeedupto10km/hr.?9. Let P= (x, y) and Q = (1, 5). Write an equation that states that the slope of line PQis 3. Show how this slope equation can be rewritten in the form y5 = 3(x1). This linearequationissaidtobeinpoint-slopeform. Explaintheterminology. FindcoordinatesforthreedierentpointsPthattthisequation.10. (Continuation) What do the lines y= 3(x1)+5, y= 2(x1)+5, and y= 12(x1)+5allhaveincommon?Howdotheydierfromeachother?11. Anotherwordchain: Turnbigintoredintowin. Changeoneletteratatime,alwaysspellingrealwords.August2012 27 PhillipsExeterAcademyMathematics11. Giventhat48 n 1296and24 d 36,whatarethelargestandsmallestvaluesthattheexpressionndcanpossiblyhave?Writeyouranswersmallest nd largest.2. Jesshas60ouncesofanalloythatis40%gold. Howmanyouncesofpuregoldmustbeaddedtothisalloytocreateanewalloythatis75%gold?3. The table at right shows data that Morgan collected during a10-milebikeridethattook50minutes. Thecumulativedistance(measuredinmiles)istabledatten-minuteintervals.(a)Makeascatterplotofthisdata. Whymightyouexpectthedatapointstolineup?Whydotheynotlineup?(b) Morgansnextbikeridelastedfor90minutes. Estimateitslength (in miles), and explain your method. What if the bike ridehadlastedtminutes;whatwoulditslengthbe,inmiles?50.040.030.020.010.00.0time10.08.25.74.42.30.0dist4. Write an equation for the line that goes through the point (1,5) and that has slope23 .5. The equation 5x8y= 20 expressesa linear relationship betweenx and y. The point(15, 7) is either on the graph of this line, above it, or below it. Which?How do you know?6. Write anequation forthe linethatcontainsthepointsinthetable,andmakeupacontextforit.x0 15 30 45 60y100 160 220 280 3407. Onanumberline,howfariseachofthefollowingnumbersfromzero?(a)45 (b) 7 (c)x (d)x + 2 (e)08. Solve(a)A =12bhforb;(b)A = 2rh + r2forh.9. Onanumberline,howfariseachofthefollowingnumbersfrom5?(a)17 (b) 4 (c)x (d)x + 3 (e)x 110. Interpreteach of the following as the distance between two numberson a number line.(a) |x 7| (b) |3 x| (c) |x + 5| (d) |x|11. Tographlinearequationssuchas3x + 5y= 30,onecanputtheequationintoslo