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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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
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
Needhelp?Visitwww.rsgsupport.org
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
