6.3 Inerpreing Sinusoidal Funcions GOAL Relae deails of sinusoidal phenomena o heir graphs. LEARN ABOUT he Mah Two sudens are riding heir bikes. A pebble is suck in he ire of each bike. The wo graphs show he heighs of he pebbles above he ground in erms of ime.? Heigh (cm) 6 5 4 3 1 h Heigh of a Pebble Bike A Bike B 1 Wha informaion abou he bikes can you gaher from he graphs of hese funcions? 3 EXAMPLE 1 Connecing he graph of a sinusoidal funcion o he siuaion Joanne s Soluion: Comparing Peaks of a Sinusoidal Funcion For Bike A, he pebble was iniially a is highes heigh of 6 cm. For Bike B, he pebble was iniially a is lowes heigh of cm. The wheels have differen diameers. The diameer of he wheel on Bike A is 6 cm. The diameer of he wheel on Bike B is 5 cm. For Bike A, he graph sars a a peak. For Bike B, he graph sars a a rough. I noiced ha he peaks on he graph are differen. The peak for Bike A is a h 5 6, which is greaer han he peak for Bike B, which is a h 5 5. The roughs, however, are he same, h 5. Glen s Soluion: Comparing Periods The wheel on Bike A akes.6 s o complee one revoluion. The wheel on Bike B akes.5 s o complee one revoluion. The period of Bike A is.6 s. The period of Bike B is.5 s. The graph for Bike A complees 5 cycles in 3 s, so he period, or lengh of one cycle, is.6 s. The graph for Bike B complees cycles in 1 s, so he period is.5 s. Chaper 6 Sinusoidal Funcions 365
Sco s Soluion: Comparing Equaions of he Axes in Sinusoidal Funcions Bike A: 6 1 Bike B: 5 1 5 3 5 5 The equaion of he axis for Bike A is h 5 3. The equaion of he axis for Bike B is h 5 5. The axle for he wheel on Bike A is 3 cm above he ground. The axle for he wheel on Bike B is 5 cm above he ground. The axis is halfway beween a peak (or maximum) and a rough (or minimum). I added he maximum and he minimum and hen divided by. Karen s Soluion: Comparing Speeds Circumference: Bike A C A 5 pr A C A 5 p(3) C A 5 6p C A 8 188.5 cm C A 8 1.885 m s A 5 d s A 5 1.885.6 s A 8 3.14 m>s Bike B C B 5 pr B C B 5 p(5) C B 5 5p C B 8 157.1 cm C B 8 1.571 m s B 5 d s B 5 1.571.5 s B 8 3.14 m>s Speed is equal o disance divided by ime, so firs I had o figure ou how far each bike ravels when he wheel complees one revoluion. This disance is he circumference. I calculaed he wo circumferences. To calculae he speed, I divided each circumference by he ime aken o complee one revoluion. The bikes are ravelling a he same speed. 366 6.3 Inerpreing Sinusoidal Funcions
Reflecing A. How would changing he speed of he bike affec he sinusoidal graph? B. For a hird rider ravelling a he same speed bu on a bike wih a larger wheel han ha on Bike A, how would he graph of he resuling sinusoidal funcion compare wih ha for Bike A and Bike B? C. Wha ype of informaion can you learn by examining he graph modelling he heigh of a pebble suck on a ire in erms of ime? APPLY he Mah EXAMPLE Comparing graphs and siuaions Annee s shop eacher was discussing able saws. The eacher produced wo differen graphs for wo differen ypes of saw. In each case, he graphs show he heigh of one ooh on he circular blade relaive o he cuing surface of he saw in erms of ime. Table Saw A Table Saw B Heigh of cuing ooh (inches) 4 6 8 1 h() Table Saw A..4.6.8.1.1 Heigh of cuing ooh (inches) 4 6 8 1 h() Table Saw B..4.6.8.1.1 Wha informaion abou he able saws can Annee gaher from he graphs? Chaper 6 Sinusoidal Funcions 367
Repko s Soluion The blade on Table Saw A is se higher han he blade on Table Saw B. The blade on Table Saw A akes. s o complee one revoluion. The peaks on he graph are differen. The peak for A is a h 5 ; he peak for B is a h 5 1. One of he easies ways o find he period is o figure ou how long i akes o go from one peak on he graph o he nex. On graph A, he firs peak is a s, and he nex is a. s. This means ha he period of graph A is. s. The blade on Table Saw B akes.3 s o complee one revoluion. The axle for he blade on Table Saw A is 3 in. below he cuing surface. The axle for he blade on Table Saw B is 5 in. below he cuing surface. The radius of he circular cuing blade on Table Saw A is 5 in. The radius of he circular cuing blade on Table Saw B is 6 in. On graph B, he firs peak is a s, and he nex is a.3 s. The period of graph B is.3 s. For graph A, I found he equaion of he axis by adding and 8 and hen dividing by. Tha gave me 3. The equaion of he axis for graph A is h 53. For graph B, I added 1 and 11 and hen divided by. Tha gave me 5. The equaion of he axis for graph B is h 55. For graph A, I go he ampliude by aking he difference beween and 3. The ampliude for graph A is 5. For graph B, he ampliude is he difference beween 1 and 5. The ampliude for graph B is 6. In boh cases, he disance from he axis o a peak represens he radius of he circular cuing blade. 368 6.3 Inerpreing Sinusoidal Funcions
EXAMPLE 3 Using echnology o undersand a siuaion The funcion j() 5 4.1 sin(64.7) 1 5.8, where is ime in years since May 199 and j() is he number of applicaions for jobs each week (in hundreds), models demand for employmen in a paricular ciy. a) Using graphing echnology in DEGREE mode and he WINDOW seings shown, graph he funcion and hen skech he graph. b) How long is he employmen cycle? Explain how you know. c) Wha is he minimum number of applicaions per week in his ciy? d) Calculae j(1), and explain wha i represens in erms of he siuaion. Karl s Soluion a) Job applicaions/ week (hundreds) 1 8 4 j() 5 1 Time (years) 15 I skeched all he cycles he window showed. b) The employmen cycle is 5.56 years, he disance beween peaks or roughs. c) The minimum number of applicaions per week is 17. d) 6.9 1.34 5 5.56 j(1) 5 1.88 There were 188 applicaions in May. To calculae he cycle, I calculaed he x-inerval beween he firs and second peak. I looked for a rough on he graph and read he j-coordinae. I looked for he place where he -coordinae was 1. In Summary Key Idea The sine and cosine funcions can be used as models o solve problems ha involve many ypes of repeiive moions and rends. Need o Know If a siuaion can be described by a sinusoidal funcion, he graph of he daa should form a series of symmerical waves ha repea a regular inervals. The ampliude of he sine or cosine funcion depends on he siuaion being modelled. One cycle of moion corresponds o one period of he sine funcion. The disance of a circular pah is calculaed from he circumference of he pah. The speed of an objec following a circular pah can be calculaed by dividing he disance by he period, he ime o complee one roaion. Chaper 6 Sinusoidal Funcions 369
CHECK Your Undersanding 1. Olivia was swinging back and forh in fron of a moion deecor when he deecor was acivaed. Her disance from he deecor in erms of ime can be modelled by he graph shown. Disance (m) 14 1 6 d() Olivia s Moion 4 8 1 16 a) Wha is he equaion of he axis, and wha does i represen in his siuaion? b) Wha is he ampliude of his funcion? c) Wha is he period of his funcion, and wha does i represen in his siuaion? d) How close did Olivia ge o he moion deecor? e) A 5 7 s, would i be safe o run beween Olivia and he moion deecor? Explain your reasoning. f) If he moion deecor was acivaed as soon as Olivia sared o swing from a res, how would he graph change? (You may draw a diagram or a skech.) Would he resuling graph be sinusoidal? Why or why no?. Marianna colleced some daa on wo paddle wheels on wo differen boas and consruced wo graphs. Analyze he graphs, and explain how he wheels differ. Refer o he radius of each wheel, he heigh of he axle relaive o he waer, he ime aken o complee one revoluion, and he speed of each wheel. Heigh of paddle (m) 6 7 5 4 3 1 1 y Paddle Wheeler A 4 Paddle Wheeler B x 48 3. Draw wo sinusoidal funcions ha have he same period and axes bu have differen ampliudes. 37 6.3 Inerpreing Sinusoidal Funcions
PRACTISING 4. Evan s eacher gave him a graph o help him undersand he speed a which a K ooh on a saw blade ravels. The graph shows he heigh of one ooh on he circular blade relaive o he cuing surface relaive o ime. a) How high above he cuing surface is he blade se? b) Wha is he period of he funcion, and wha does i represen in his siuaion? c) Wha is he ampliude of he funcion, and wha does i represen in his siuaion? d) How fas is a ooh on he circular cuing blade ravelling in inches per second? 5. An oscilloscope hooked up o an alernaing curren (AC) circui shows a sine curve on is display. a) Wha is he period of he funcion? Include he unis of measure. b) Wha is he equaion of he axis of he funcion? Include he unis of measure. c) Wha is he ampliude of he funcion? Include he unis of measure. 6. Skech a heigh-versus-ime graph of he sinusoidal funcion ha models each siuaion. Draw a leas hree cycles. Assume ha he firs poin ploed on each graph is a he lowes possible heigh. a) A Ferris wheel wih a radius of 7 m, whose axle is 8 m above he ground, and ha roaes once every 4 s b) A waer wheel wih a radius of 3 m, whose cenre is a waer level, and ha roaes once every 15 s c) A bicycle ire wih a radius of 4 cm and ha roaes once every s d) A girl lying on an air maress in a wave pool ha is 3 m deep, wih waves.5 m in heigh ha occur a 7 s inervals 7. The ables show he varying lengh of dayligh for Timmins, Onario, locaed a a laiude of 48, and Miami, Florida, locaed a a laiude of 5. The lengh of he day is calculaed as he inerval beween sunrise and sunse. Heigh (in.) Curren (amperes) Heigh of a Saw Tooh h().4.8.1 4 6 8 1 4.5 3. 1.5 1.5 3. 4.5 Oscilloscope Display y x.4.8 Timmins, a laiude 488 Day of Year 15 46 74 15 135 165 196 7 58 88 319 349 Hours of Dayligh 8.8 1. 11.9 13.7 15. 16.1 15.7 14.4 1.6 1.9 9. 8.3 Miami, a laiude 58 Day of Year 15 46 74 15 135 165 196 7 58 88 319 349 Hours of Dayligh 1.7 11.3 1. 1.8 13.6 13.8 13.6 13.1 1.3 11.6 1.9 1.5 Chaper 6 Sinusoidal Funcions 371
a) Plo he daa on separae coordinae sysems, and draw a smooh curve hrough each se of poins. b) Compare he wo curves. Refer o he periods, ampliudes, and equaions of he axes. c) Wha migh you infer abou he relaionship beween hours of dayligh and he laiude a which you live? 8. The diameer of a car s ire is 5 cm. While he car is being driven, he ire T picks up a nail. a) Draw a graph of he heigh of he nail above he ground in erms of he disance he car has ravelled since he ire picked up he nail. b) How high above he ground will he nail be afer he car has ravelled.1 km? c) How far will he car have ravelled when he nail reaches a heigh of cm above he ground for he fifh ime? d) Wha assumpion mus you make concerning he driver s habis for he funcion o give an accurae heigh? 9. In high winds, he op of a signpos vibraes back and forh. The disance he ip of he pos vibraes o he lef and righ of is resing posiion can be defined by he funcion d() 5 3 sin(18), where d() represens he disance in cenimeres a ime seconds. If he wind speed decreases by km/h, he vibraion of he ip can be modelled by he funcion d() 5 sin(18). Using graphing echnology in DEGREE mode and he WINDOW seings shown, produce he wo graphs. How does he reduced wind speed affec he period, ampliude, and equaion of he axis? 1. The heigh, h(), of a baske on a waer wheel a ime can be modelled by h() 5 sin(1) 1 1.5, where is in seconds and h() is in meres. a) Using graphing echnology in DEGREE mode and he WINDOW seings shown, graph h() and skech he graph. b) How long does i ake for he wheel o make a complee revoluion? Explain how you know. c) Wha is he radius of he wheel? Explain how you know. d) Where is he cenre of he wheel locaed in erms of he waer level? Explain how you know. e) Calculae h(1), and explain wha i represens in erms of he siuaion. 11. The equaion h() 5.5 sin(7) models he displacemen of a buoy in A meres a seconds. a) Using graphing echnology in DEGREE mode and he WINDOW seings shown, graph h() and skech he graph. b) How long does i ake for he buoy o ravel from he peak of a wave o he nex peak? Explain how you know. c) How many waves will cause he buoy o rise and fall in 1 min? Explain how you know. d) How far does he buoy drop from is highes poin o is lowes poin? Explain how you know. 37 6.3 Inerpreing Sinusoidal Funcions
1. The average monhly emperaure, T(), in degrees Celsius in Kingson, Onario, can be modelled by he funcion T() 5 14. sin(3( 4.)) 1 5.9, where represens he number of monhs. For 5 1, he monh is January; for 5, he monh is February; and so on. a) Using graphing echnology in DEGREE mode and he WINDOW seings shown, graph T() and skech he graph. b) Wha does he period represen in his siuaion? c) Wha is he average emperaure range in Kingson? d) Wha is he mean emperaure in Kingson? e) Calculae T(3), and explain wha i represens in erms of he siuaion. 13. Two wrecking balls aached o differen cranes swing back and forh. The disance he balls move o he lef and he righ of heir resing posiions in erms of ime can be modelled by he graphs shown. Disance (m) 4 4 d() Ball A Ball B 8 16 4 a) Wha is he period of each funcion, and wha does i represen in his siuaion? b) Wha is he equaion of he axis of each funcion, and wha does i represen in his siuaion? c) Wha is he ampliude of each funcion, and wha does i represen in his siuaion? d) Deermine he range of each funcion. e) Compare he moions of he wo wrecking balls. 14. How many pieces of informaion do you need o know o skech a sinusoidal C funcion. Wha pieces of informaion could hey be? Exending 15. A gear of radius 1 m urns counerclockwise and drives a larger gear of radius 4 m. Boh gears have heir axes along he horizonal. a) In which direcion is he larger gear urning? b) If he period of he smaller gear is s, wha is he period of he larger gear? c) In a able, record convenien inervals for each gear, o show he verical displacemen, d, of he poin where he wo gears firs ouched. Begin he able a s and end i a 4 s. Graph verical displacemen versus ime. d) Wha is he displacemen of he poin on he large wheel when he drive wheel firs has a displacemen of.5 m? e) Wha is he displacemen of he drive wheel when he large wheel firs has a displacemen of m? f) Wha is he displacemen of he poin on he large wheel a 5 min? Chaper 6 Sinusoidal Funcions 373