6-6 Modeling Real-World Data with Sinusoidal Functions

1 Lesson 6-6 Modeling Real-World Data with Sinusoidal Functions387 Modeling Real-World Datawith Sinusoidal FunctionsMETEOROLOGYThe table contains the times that the sun rises and sets on the fifteenth of every month in Brownsville, t 1 represent January t 2 represent February t 3 represent March 15. Write a function that models the hours of daylight for Brownsville. Use your model to estimate the number of hours of daylight on September 30. This problem will be solved in Example you can determine the function for the daylight, you must firstcompute the amount of daylight for each day as a decimal value. ConsiderJanuary 15. First, write each time in 24-hour :19 7:196:00 6:00 12 or 18:00 Then change each time to a decimal rounded to the nearest :19 7 1690 or :00 18 600 or January 15, there will be or hours of , the number of daylight hours can be determined for the fifteenth ofeach Model Real-World data using sineand cosinefunctions. Use sinusoidalfunctions tosolve :196:00 February7:056:23 March6:406:39 April6:076:53 May5:447:09 June5:387:23 July5:487:24 August6:037:06 September6:166:34 October6:296:03 November6:485:41 December7:095 of of there are 12 months in a year,month 13 is the same as month 1, month 14 is the same as month 2, and so on.

2 The function is periodic. Enter the data into a graphing calculator and graph thepoints. The graph resembles a type of sinecurve. You can write a Sinusoidal functionto represent the data. A Sinusoidal functioncan be any function of the form y Asin (k c) h or y Acos (k c) to the application at the beginning of the Write a function that models the amount of daylight for Use your model to estimate the number of hours of daylight onSeptember data can be modeled by a function of the form y Asin (kt c) h,where tis the time in months. First, find A, h, and :A or :h or : 2k 12 The period is 6 Substitute these values into the general form of the Sinusoidal Asin (kt c) hy 6 t c , k 6 , h compute c, substitute one of the coordinate pairs into the sin 6 t c sin 6 (1) c (t, y) (1, ). sin 6 c Add to each side. sin 6 c Divide each side by 1 6 cDefinition of inversesin 1 6 cAdd 6 to each side. cUse a 6 Graphs of Trigonometric Functions [ 1, 13] scl:1 by [ 1, 14] scl:1 ResearchFor data aboutamount of daylight, average temperatures,or tides, is half the difference between the mostdaylight ( h) and the least daylight( h).

3 H is half the sum of the greatest value andleast function y sin 6 t is one model for the daylightin check this answer, enter the datainto a graphing calculator andcalculate the SinRegstatistics. Rounding to the nearest hundredth, y sin ( ) The models are similar. Either modelcould be 30 is half a month past September 15, so t Select a modeland use a calculator to evaluate it for t 1: Paper and Pencily sin 6 t sin 6 ( ) 2: Graphing Calculatory sin ( ) sin [ ( ) ] September 30, Brownsville will have about hours of general, any Sinusoidal function can be written as a sine function or as acosine function. The amplitude, the period, and the midline will remain the , the phase shift will be different. To avoid a greater phase shift thannecessary, you may wish to use a sine function if the function is about zero at x 0 and a cosine function if the function is about the maximum or minimum at x average seated adult breathes in and out every 4 seconds.

4 Theaverage minimum amount of air in the lungs is liter, and the averagemaximum amount of air in the lungs is liter. Suppose the lungs have aminimum amount of air at t 0, where tis the time in Write a function that models the amount of air in the Graph the Determine the amount of air in the lungs at 6-6 Modeling Real-World Data with Sinusoidal Functions389 GraphingCalculatorTipFor keystroke instruction on how tofind sine regression statistics, see page A25.(continued on the next page) the function has its minimum value at t 0, use the cosine function. A cosine function with its minimum value at t 0 has no phase shift and a negative value for A. Therefore, the general form of the model is y Acos kt h, where tis the time in seconds. Find A, k, and :A or :h or : 2k 4 The period is 2 Therefore, y cos 2 t models the amount of air in the lungs of an average seated a graphing calculator to graph the this function to find the amount of air in the lungs at cos 2 t cos 2 ( ) lungs have about liter of air at and study the lesson to answer each Sinusoidal function in your own and contrast Real-World data that can be modeled with a polynomial function and Real-World data that can be modeled with a Sinusoidal three Real-World examples that can be modeled with a sinusoidalfunction.

5 390 Chapter 6 Graphs of Trigonometric FunctionsA is half the difference between the greatestvalue and the least is half the sum of the greatest value and theleast value.[ 2, 10] scl:1 by [ , 1] the equilibrium point is y 0, then y 5 cos 6 t models a buoybobbing up and down in the the location of the buoy when t is the maximum height of the buoy? the location of the buoy at t certain person s blood pressure oscillates between 140 and 80. If theheart beats once every second, write a sine function that models the person sblood average monthly temperatures for the city of Seattle,Washington, are given the amplitude of a Sinusoidal function that models the the vertical shift of a Sinusoidal function that models the is the period of a Sinusoidal function that models the monthlytemperatures? a Sinusoidal function that models the monthly temperatures, using t 1 to represent to your model, what is the average monthly temperature inFebruary?

6 How does this compare to the actual average? to your model, what is the average monthly temperature inOctober? How does this compare to the actual average? initial behavior of the vibrations of the note E above middle C canbe modeled by y sin 660 is the amplitude of this model? is the period of this model? the frequency (cycles per second) for this rodeo performer spins a lasso in a circle perpendicular to the ground. The height of the knot from the ground is modeled by h 3 cos 53 t , where tis the time measured in is the highest point reached by the knot? is the lowest point reached by the knot? is the period of the model? to the model, find the height of the knot after 25 PracticeApplicationsand ProblemSolvingLesson 6-6 Modeling Real-World Data with Sinusoidal March April May June July Aug. Sept. Oct. Nov. 44 47 50 56 61 65 66 61 54 46 42 a certain region with hawks as predators and rodents as prey, the rodent population Rvaries according to the model R 1200 300 sin 2 t , and the hawk population Hvaries according to the model H 250 25 sin 2 t 4 , with tmeasured in years since January 1, was the population of rodents on January 1, 1970?

7 Was the population of hawks on January 1, 1970? are the maximum populations of rodents and hawks? Do these maximaever occur at the same time? what date was the first maximum population of rodents achieved? is the minimum population of hawks? On what date was the minimumpopulation of hawks first achieved? to the models, what was the population of rodents and hawks onJanuary 1 of the present year? leaf floats on the water bobbing up and down. The distance betweenits highest and lowest point is 4 centimeters. It moves from its highest pointdown to its lowest point and back to its highest point every 10 seconds. Write acosine function that models the movement of the leaf in relationship to theequilibrium a sine function which models the oscillation of tides in Savannah,Georgia, if the equilibrium point is feet, the amplitude is feet, the phaseshift is hours, and the period is mean average temperature in Buffalo, New York, is .The temperature fluctuates above and below the mean temperature.

8 If t 1 represents January, the phase shift of the sine function is a model for the average monthly temperature in to your model, what is the average temperature in March? to your model, what is the average temperature in August?392 Chapter 6 Graphs of Trigonometric average monthly temperatures for the city of Honolulu,Hawaii, are given the amplitude of a Sinusoidal function that models the the vertical shift of a Sinusoidal function that models the is the period of a Sinusoidal function that models the monthlytemperatures? a Sinusoidal function that models the monthly temperatures, using t 1 to represent to your model, what is the average temperature in August? Howdoes this compare to the actual average? to your model, what is the average temperature in May? How doesthis compare to the actual average? ThinkingWrite a cosine function that is equivalent to y 3 sin (x ) Head in Nova Scotia, Canada, is known for its extremefluctuations in tides.

9 One day in April, the first high tide rose to feet at 4:30 The first low tide at feet occurred at 10:51 The second hightide was recorded at 4:53 the amplitude of a Sinusoidal function that models the the vertical shift of a Sinusoidal function that models the is the period of a Sinusoidal function that models the tides? a Sinusoidal function to model the tides, using tto represent thenumber of hours in decimals since to your model, determine the height of the water at 7:30 table at the rightcontains the times that the sun risesand sets in the middle of each monthin New York City, New York. Supposethe number 1 represents the middleof January, the number 2 representsthe middle of February, and so the amount of daylight hoursfor the middle of each is the amplitude of asinusoidal function that models thedaylight hours? is the vertical shift of asinusoidal function that models thedaylight hours? is the period of a sinusoidalfunction that models the daylighthours?

10 A Sinusoidal function that models the daylight 6-6 Modeling Real-World Data with Sinusoidal March April May June July Aug. Sept. Oct. Nov. 73 74 76 78 79 81 81 81 80 77 74 :194:47 February6:565:24 March6:165 :29 May4:447:01 June4:247:26 July4:337:28 August5:017:01 September5:316:14 October6:015:24 November6:364:43 December7:084:28 Mixed ThinkingThe average monthly temperature for Phoenix, Arizona can be modeled by y sin 6 t c . If the coldest temperature occurs in January (t 1), find the value of years ago, an amusement park in Sandusky, Ohio,had a ride called the Rotor in which riders stood against the walls of aspinning cylinder. As the cylinder spun, the floor of the ride dropped out, andthe riders were held against the wall by the force of friction. The cylinder ofthe Rotor had a radius of meters and rotated counterclockwise at a rate of14 revolutions per minute. Suppose the center of rotation of the Rotor was atthe origin of a rectangular coordinate the initial coordinates of the hinges on the door of the cylinder are (0, ), write a function that models the position of the door at the coordinates of the hinges on the door at 4 an alternating current,the instantaneous voltage VRisgraphed at the right.