OnFebruary11,2008,SingaporeopenedanewobservationwheelcalledtheSingapore Flyer.Atthetimeofitsopening,thisgiantFerriswheelwasthetallestintheworld.The SingaporeFlyerconsistsofanobservationwheelwithadiameterof150metersatop aboardingterminal,givingthestructureanoverallheightof165meters.Twenty-eightair-conditionedcapsules rotate onthe outsideofthe wheel toprovideunobstructedviewsofthecity. The wheelrotates at a constant rate of 26 centimeters per second. Thisis slowenoughthatthewheeldoesnotneedtostopforloadingandunloadingunlessthereare specialpassengerneeds.

1.Usinggraphpaper,drawanaccuratediagramofthewheelshowingthedimensionsgivenabove.Use acompass or othertooltoaccuratedrawthe circle.

2.Onthenextpage,fillinthetableshowingtheheightofasinglecapsulechangingasit rotatescounterclockwisefromtheboardingterminalaroundthewheel.Todothis, calculate thecircumference of the wheel.

a.Howmanyminutesdoesittakea capsule tomakeone completerevolution aroundthewheel?(Roundtothenearestminute.)Explainyourprocess.

b.Beforecompletingthetable,explainhowtheanglevaluesprovidedinthetableare correct.

c.Thefirstinscribedanglethatmodelsthesituationafteroneminuteisshown.Useadditionaldiagramsandinscribedrighttrianglestodeterminemorevaluesofthe totalheightasagivencapsulecontinuestorotatethroughonecompleterevolution. Usetrigonometrytocalculatethecorrespondingvaluesofaandcompletethetable,findingtheheightofthecapsuleatthevariousintervalsoftime.

Time (min) / AngleofRotationfromBoardingStation / VerticalLegofRightTriangle(a) / Process
(Ifyoudecidetochangeyourprocess,explainyourdecision.) / TotalHeight ofCapsule(m)
1 / 12°
30 / 360°

3.Createagraphshowingtheheightchangingasagivencapsulerotatesthroughone completerevolutionofthewheel.Showat least10 well-spaced datapoints on yourgraph.

4.Usespreadsheetsoftwareoragraphingcalculatortomodeltheheightofacapsuleasit continuestorotatearoundthewheel.Showatleast90minutesofrotation.Createa graphofthedata.

5.Onyourgraph,labelthe periodand amplitudeofthecurve.Howdothese valuesonyourmathematicalmodelrelatetothephysicalcontextoftheSingaporeFlyer?

6.REFLECTION:Inthisproblem,yourclassmatesuseddifferentmethodstosolvethisproblem.Nowthatyouhaveseenthedifferentprocesses, whichweremostuseful?Ifyouhadtodothisproblemusingadifferentprocessthantheoneyouused,whichmethod wouldyouchoose?Whatinformationwould youneed toknow? Whatcalculations wouldyouhavetododifferently?Howweretrigonometricratiosusedinthedifferentprocesses??

7.EXTENSION

  • ResearchanothergiganticFerriswheel.Findthedimensionsofthewheelandhowlongittakesforthewheeltocompleteonerevolution.Usingthatinformation,makearoughsketchoftheheightofacapsuleover time.

OR

  • Whatother situations canyouthink ofthat repeatthemselvesandcould possiblybemodeledwithaperiodicfunction?(Someexamplesincludevolumeofairinlungsovertimeduringrest,a pendulumswinging,thebounceofaspring, sealevelsaffectedbytides, and sound waves.)