Chapter Review, page 581

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1 = 360 = 360 = 80 Horizontal translation = 60 to the right d 4 The range of this graph is {y 6 y, y R}, and its amplitude is. The is 80 and the horizontal translation is 60 to the right. The equation of the midline is To determine the equation of the sinusoidal regression function, plot the data using a graphing calculator and use the data to determine the equation. The equation of the sinusoidal regression function for this data is sin (0.476 x.76 ) Chapter Review, page 58. a) e.g., 5 is one quarter of is aout radian. One quarter of radian is 0.5 radians. It is aout 0.3 radians. 6. a) e.g., 3 is slightly less than 80 60, or is aout 3. radians. 60 is aout radian. 3. =. 0 is aout. radians. It is aout. radians. c) function for this data is.75 sin (.570 x). Period = second maximum first maximum Period = 5 Period = 4 The of this graph is 4 s. c) e.g., Positive and negative velocities correspond to exhalations and inhalations (or vice versa). e) The velocity of air eing 0 L/s corresponds to the x-intercepts of the graph. Between 9 s and 9 s, the velocity of the air will e 0 at 0 s, s, 4 s, 6 s, and 8 s. e.g., 0 is slightly greater than , or is aout 3. radians. 30 is one half of is aout radian. One half of radian is 0.5 radians = is aout 3.7 radians. It is aout 3.9 radians. Foundations of Mathematics Solutions Manual 8-3

2 d) c) e.g., 555 = is aout 6.3 radians. 80 is aout 3. radians. 5 is one quarter of is aout radian. One quarter of radian is 0.5 radians = 9.75 It is aout 9.8 radians. e.g., 0.5 radians is one half of radian. radian is aout 60. One half of 60 is radians is aout 30. d). a) e.g., 9.5 = radians is aout = radians is aout 540. e.g., 5.4 radians is slightly less than 6.3, or 5.3 radians. radian is aout = radians is aout radians is aout 90. e.g.,.0 = radians is aout = radians is aout a) The values of x where sin x < 0 over the interval from 0 to 70 are {x 80 < x < 360, x R} or { x 540 < x < 70, x R}. The values of x where cos x < 0 over the interval from 0 to 70 are {x 0 x < 90, x R} or { x 70 < x < 450, x R} or { x 630 < x 70, x R}. 4. a) Range: {y 4 y 0, y R} Maximum = d + a Minimum = d a Maximum = + Minimum = Maximum = 0 Minimum = 4 max min 0 ( 4) Chapter 8: Sinusoidal Functions

3 d The range of this graph is {y 4 y 0, y R}, and its amplitude is. The equation of the midline is, and the graph s is 360. Range: {y.5 y 0.5, y R} Maximum = d + a Maximum =.5 + Maximum = 0.5 Minimum = d a Minimum =.5 Minimum =.5 max min 0.5 (.5) max+ min (.5) 3.5 The range of this graph is {y.5 y 0.5, y R}, and its amplitude is. The equation of the midline is.5, and the graph s is a) Maximum = 4 m Minimum = m The equation of the midline is 8 m. The midline represents the position of the swing at rest or her initial distance from the motion sensor. She started at 8 m from the sensor. 4 6 The amplitude of the function is 6 m. c) Period = second max first max Period = 7 3 Period = 4 s The of the function is 4 seconds. The represents the time it takes to swing ack and forth once. d) The closest Olivia came to the motion detector is m. e) e.g., Yes, since she is at her furthest distance from the detector. t = 7 s corresponds to a maximum, so Olivia is her maximum distance, 4 m, away from the motion detector. 6. The radius of each wheel corresponds to the amplitude of the graph. The height of the axle relative to the water corresponds to the equation of the midline. The time taken to complete one revolution corresponds to the of the graph. a) Paddle Wheeler A: Maximum = 5 m Minimum = m 5 ( ) ( ) 4 Foundations of Mathematics Solutions Manual 8-5

4 Period: The curve repeats at aout 3.5 squares on the graph, which is aout = s. 7 Period = s The radius of the wheel is 3 m, the height of the axle relative to the water is m, and the time taken to complete one revolution is s. Paddle Wheeler B: Maximum = 7 m Minimum = m 7 ( ) ( ) 6 3 Period: The curve repeats at aout 4.7 squares on the graph, which is aout = 6 s. 7 Period = 6 s The radius of the wheel is 4 m, the height of the axle relative to the water is 3 m, and the time taken to complete one revolution is 6 s. They are travelling at aout the same speed (75 m in 48 s). 7. a) The siren has a higher frequency as it drives towards you, since the is shorter. Towards you: π π Away from you π π The frequency as the emergency vehicle drives towards you and away from you is 877 Hz and 735 Hz respectively. 8. a) 3 ( 7) 0 5 The amplitude of this graph is 5, which eliminates equation i). The rest of the equations are equivalent except for the horizontal translations. Graph A crosses the midline and is increasing at 30, so the horizontal translation of the sine function is 30 to the right. Equation iv) corresponds to Graph A. The amplitude of this graph is the same as Graph A, a = 5, which eliminates equation i). The first maximum of Graph B occurs at 30, so the horizontal translation of the cosine function is 30 to the right. Equation ii) corresponds to Graph B. 9. a) a = 4, = 6, c =.5, d = 3 Maximum = d + a Maximum = Maximum = 7 Minimum = d a Minimum = 3 4 Minimum = 8-6 Chapter 8: Sinusoidal Functions

5 7 ( ) ( ) 6 3 Range: {y y 7, y R} Period = π Period = π 6 Period = π 3 Horizontal translation =.5 to the right The amplitude of this function is 4. The equation of the midline is 3. The range of this equation is {y y 7, y R}. The is π 3 or.047. The horizontal translation is.5 to the right. 0. π π 5.08 π π π 3.4 π The frequency of the WiFi signals are.4 cycles per nanosecond and 5.0 cycles per nanosecond respectively.. e.g., To predict the average temperature in Humoldt on Decemer 8, plot the data using graphing technology and determine the equation of the sinusoidal regression function. function is sin (0.466 x.688 ) Using the data points Jan =, Fe =, etc., and assuming that the average temperature is for the middle of the month, Decemer 8 would e.75. Sustitute x = sin (0.466 x.688 ) e.g., I predict the average temperature on Decemer 8 to e aout. C.. e.g., To predict the monthly low temperatures, assume that the average temperatures occur in the middle of the month. and each day numer is for those days. Plot the data using graphing technology and determine the sinusoidal regression functions. function for Edmonton is 5.8 sin (0.05 x.453 ) 5.4. function for Rio de Janeiro is.567 sin (0.07 x +.00 ) Sustitute x = 79 (June 8) Edmonton: 5.8 sin (0.05 x.453 ) Rio de Janeiro:.567 sin (0.07 x +.00 ) Difference: = The difference is aout 9.6 C. Foundations of Mathematics Solutions Manual 8-7