SECONDARY MATH III // MODULE 6

MODELING PERIODIC BEHAVIOR – 6.5

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6.5 Moving Shadows

A Practice Understanding Task

Inspiteofhisnervousness,CarlosenjoyshisfirstrideontheamusementparkFerris

wheel.Hedoes,however,spendmuchofhistimewithhiseyesfixedonthegroundbelowhim.

Afterawhile,hebecomesfascinatedwiththefactthatsincethesunisdirectlyoverhead,his

shadowmovesbackandforthacrossthegroundbeneathhimasheridesaroundontheFerris

wheel.

RecallthefollowingfactsfortheFerriswheelCarlosisriding:

• TheFerriswheelhasaradiusof25feet• ThecenteroftheFerriswheelis30feetabovetheground• TheFerriswheelmakesonecompleterotationcounterclockwiseevery20seconds

TodescribethelocationofCarlos’shadowasitmovesbackandforthonthegroundbeneath

him,wecouldmeasuretheshadow’shorizontaldistance(infeet)totherightorleftofthepoint

directlybeneaththecenteroftheFerriswheel,withlocationstotherightofthecenterhaving

positivevalueandlocationstotheleftofthecenterhavingnegativevalues.Forinstance,inthis

systemCarlos’shadow’slocationwillhaveavalueof25whenheisatthepositionfarthesttothe

rightontheFerriswheel,andavalueof-25whenheisatapositionfarthesttotheleft.

1. WhatwouldCarlos’positionbeontheFerriswheelwhenhisshadowislocatedat0inthisnewmeasurementsystem?

2. Sketchagraphofthehorizontallocationof

Carlos’shadowasafunctionoftimet,wheretrepresentstheelapsedtimeafterCarlospassespositionA,thefarthestrightpositionontheFerriswheel.

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24

SECONDARY MATH III // MODULE 6

MODELING PERIODIC BEHAVIOR – 6.5

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

3. CalculatethelocationofCarlos’shadowatthetimestgiveninthefollowingtable,wheretrepresentsthenumberofsecondssinceCarlospassedthepositionfarthesttotherightontheFerriswheel.Keeptrackofanyregularitiesyounoticeinthewaysyoucalculatethelocationoftheshadow.Asyoucalculateeachlocation,plotCarlos’positiononthediagramattheright.

ElapsedtimesincepassingpositionA

CalculationsHorizontalposition

oftherider

1sec

2sec

3sec

4sec

8sec

14sec

18sec

23sec

28sec

40sec

4. Writeageneralformulaforfindingthelocationoftheshadowatanyinstantintime.

25

SECONDARY MATH III // MODULE 6

MODELING PERIODIC BEHAVIOR – 6.5

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

6.5

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READY Topic:Comparingradiusandarclength

Stagecoachesandwagonsinthe1800shadwheelsthatweresmallerinthefrontthanintheback.

Thefrontwheelshad12spokes.Thetopofthefrontwheelmeasured44inchesfromtheground.

Therearwheelshad16spokes.Thetopoftherearwheelmeasured59inchesfromtheground.

(Fortheseproblemsdisregardthehubatthecenterofthewheel.Assumethespokesmeetinthe

centeratapoint.Leaveπinyouranswers.)

1. Findtheareaandthecircumferenceofeachwheel.

2. Determinethecentralanglebetweenthespokesoneachwheel.

3. Findthedistanceonthecircumferencebetweentwoconsecutivespokesforeachwheel.

4. Thewagonscouldcoveradistanceof15milesperday.Howmanymoretimeswouldthefront

wheelturnthanthebackwheelonanaverageday?

5. Awheelrotatesrtimesperminute.Writeaformulaforhowmanydegreesitrotatesint

seconds.

READY, SET, GO! Name PeriodDate

26

SECONDARY MATH III // MODULE 6

MODELING PERIODIC BEHAVIOR – 6.5

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6.5

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SET Topic:Determiningvaluesofcosineinacircle

Usethegivenpointonthecircletofindthevalueofcosine.Recall! = #$% + '%()*,-./ =

0

1.

6.

7.

8.

9.

10.Ineachgraph,theangleofrotationisindicatedbyanarcand/.Describetheanglesofrotation

thatmakethex-valuesofthepointspositiveandtheanglesofrotationthatmakethex-values

negative.

11.Whatdoyounoticeaboutthex-valuesandthevalueofcosineinthegraphs?

3−5,5√89

:−√2

2,−√2

2<

36√3,−69

27

SECONDARY MATH III // MODULE 6

MODELING PERIODIC BEHAVIOR – 6.5

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

6.5

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12.Inthegraphattheright,theradiusofthecircleis1.

Theintersectionsofthecircleandtheaxesarelabeled.

Basedonyourobservationin#11,whatdoyouthink

thevalueofcosinemightbeforthefollowingvaluesof/:

90°? 180°? 270°? 360°?

GO

Topic:Reviewingthemeasurementsinspecialtriangles

Usethegiveninformationtofindthemissingsidesandthemissingangles.

TriangleABCisarighttriangle.AngleCistherightangle.Writetheexactvaluesforthesides.

13.

14.

15.

16.

5 in 7√2 cm

13√2 m

40cd

28

SECONDARY MATH III // MODULE 6

MODELING PERIODIC BEHAVIOR – 6.5

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

6.5

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17.

18.

19.

20.

FindADinthefiguresbelow.

21. 22.

Rememberthatπissimplyanumber.

23.Ifyoupurchasedπgallonsofgasoline,abouthowmanygallonsofgasdidyoubuy?

24.Ifyouwerepaid5πdollarsperhour,abouthowmanydollarswouldyoumakein8hours?

25.Ifyouslept2πhourseachnight,abouthowmanyhoursofsleepwouldyougetpernight.

30° 14 cm

60° 25f

60°

9√3

ft

30° 12 in

1

2

i

n 1

6

f

t

11 m 16 ft

29