6.2 Sinusoidal Functions - Textbook 2 [Questions & Answers]

CHECK Your Understanding 1. Olivia was swinging back and forth in front of a motion detector when the detector was activated. Her distance from the detector in terms of time can b modelled by the graph shown. a) What is the equation of the axis, and what does it represent in this situation? b) What is the amplitude of this function? c) What is the period of this function, and what does it represent in this situation? d) How close did Olivia get to the motion detector? e) At t = 7 s, would it be safe to run between Olivia and the motion detector? Explain your reasoning. f) If the motion detector was activated as soon as Olivia started to swing from at rest, how would the graph change? (You may draw a diagram or a sketch.) Would the resulting graph be sinusoidal? Why or why not? 2. Marianna collected some data on two paddle wheels on two different boats and constructed two graphs. Analyze the graphs, and explain how the wheels differ. Refer to the radius of each wheel, the height of the axle relative to the water, the time taken to complete one revolution, and the speed of each wheel. 3. Draw two sinusoidal functions that have the same period and axes but have different amplitudes. PRACTISING 4. Evan's teacher gave him a graph to help him understand the speed at which a K tooth on a saw blade travels. The graph shows the height of one tooth on the circular blade relative to the cutting surface relative to time. How high above the cutting surface is the blade set? b) What is the period of the function, and what does it represent in this situation? c) What is the amplitude of the function, and what does it represent in this situation? d) How fast is a tooth on the circular cutting blade travelling in inches per second? 5. An oscilloscope hooked up to an alternating current (AC) circuit shows a sine curve on its display: a) What is the period of the function? Include the units of measure. b) What is the cquation of the axis of the function? Include the units of Measule c) What is the amplitude of the function? Include the units of measure. 6. Sketch a height-versus-time graph of the sinusoidal function that models each situation. Draw at least three cycles. Assume that the first point plotted on each graph is at the lowest possible height. a) A Ferris wheel with a radius of 7 m, whose axle is 8 m above the ground, and that rotates once every 40 s b) A water wheel with a radius of 3 m, whose centre is at water level, and that rotates once every 15 s c) A bicycle tire wich a radius of 40 cm and that rotates once every 2 s d) A girl lying on an air mactress in a wave pool that is 3 m deep, with waves 0.5 m in height that occur at 7 s intervals 7. The tables show the varying length of daylight for Timmins, Ontario, located at a latitude of 48°, and Miami, Florida, located at a latitude of 25°. The length of the day is calculated as the interval between sunrise and sunset. a) Plot the data on separate coordinate systems, and draw a smooth curve through each set of points. b) Compare the two curves. Refer to the periods, amplitudes, and equations of the axes. c) What might you infer about the relationship between hours of daylight and the latitude ar which you live? 8. The diameter of a car's tire is 52 cm. While the car is being driven, the tire picks up a nail. a) Draw a graph of the height of the nail above the ground in terms of the distance the car has travelled since the tire picked up the nail. b) How high above the ground will the nail be after the car has travelled 0.1 km? c) How far will the car have travelled when the nail reaches a height of 20 cm above the ground for the fifth time? d) What assumption must you make concerning the driver's habits for the function to give an accurate height? 9. In high winds, the cop of a signpost vibrates back and forth. The distance the tip of the post vibrates to the left and right of its resting position can be defined by the function d(s) = 3 sin (1080r)°, where d(r) represents the distance in centimeters at time & seconds. If the wind speed decreases by 20 km/h, the vibration of the tip can be modelled by the function d(r) = 2 sin (1080:)°. Using graphing technology in DEGREE mode and the WINDOW settings shown, produce the nwo graphs. How does the reduced wind speed affect the period, amplitude, and equation of the axis? 10. The height, h(t), of a basket on a water wheel at time t can be modelled by (t) = 2 sin(12t)° + 1.5, where z is in seconds and b (t) is in metres. a) Using graphing technology in DEGREE mode and the WINDOW settings shown, graph hs) and sketch the graph. b) How long does it take for the wheel to make a complete revolution? Explain how you know. c) What is the radius of the wheel? Explain how you know. d) Where is the centre of the wheel located in terms of the water level? Explain how you know. e) Calculate /(10), and explain what it represents in terms of the situation. 11. The equation h (c) = 2.5 sin (72t)° models the displacement of a buoy in metres at t seconds. a) Using graphing technology in DEGREE mode and the WINDOW settings shown, graph h(t) and sketch the graph. b) How long does it take for the buoy to travel from the peak of a wave to the next peak? Explain how you know. c) How many waves will cause the buoy to rise and fall in 1 min? Explain how you know. d) How far does the buoy drop from its highest point to its lowest point? Explain how you know. 12. The average monthly temperature, T(*), in degrees Celsius in Kingston, Ontario, can be modelled by the function T(t) = 14.2 sin (30 (t - 4.2))° + 5.9, where / represents the number of months. For t = 1, the month is January; for t = 2, the month is February; and so on. a) Using graphing technology in DEGREE mode and the WINDOW settings shown, graph 7(r) and sketch the graph. b) What does the period represent in this situation? c) What is the average temperature range in Kingston? d) What is the mean temperature in Kingston? e) Calculate T(30), and explain what it represents in terms of the situation. 13. Two wrecking balls attached to different cranes swing back and forth. The distance the balls move to the left and the right of their resting positions in terms of time can be modelled by the graphs shown. a) What is the period of each function, and what does it represent in this situation? b) What is the equation of the axis of each function, and what does it represent in this situation? c) What is the amplitude of each function, and what does it represent in this situation? d) Determine the range of each function. e) Compare the motions of the two wrecking balls. 14. How many pieces of information do you need to know to sketch a sinusoidal c function. What pieces of information could they be? Extending 15. A gear of radius 1 m turns counterclockwise and drives a larger gear of radius 4 m. Both gears have their axes along the horizontal. a) In which direction is the larger gear turning? b) If the period of the smaller gear is 2 s, what is the period of the larger gear? In a table, record convenient intervals for each gear, to show the vertical displacement, d, of the point where the two gears first touched. Begin the table at 0 s and end it at 24 s. Graph vertical displacement versus time. d) What is the displacement of the point on the large wheel when the drive wheel first has a displacement of - 0.5 m? e) What is the displacement of the drive wheel when the large wheel first has a displacement of 2 m? f) What is the displacement of the point on the large wheel at 5 min? a) The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is 2 and the maximum is 14, the (14 + 2)/2 equation of the axis is y = - or y = 8. In this case it represents the resting position of the swing. b) The amplitude is half the distance between the maximum and minimum values. Since the minimum is 2 and the maximum is 14, the amplitude is 14 - 2 - or 6 m. c) The period is the time it takes the function to go through one complete cycle. Since the function goes through one complete cycle between 0 s and 4 s, for example, the period is 4 - 0 or 4 s. In this case it represents the time to complete one full swing. d) From the graph, the minimum distance between Olivia and the motion detector was 2 m, so the closest Olivia got to the motion detector was 2 m. e) At 1 = 7 s, the distance between Olivia and the motion detector is 14 m. However, over the course of the next 2 s, that distance will decrease to 2 m. Therefore, whether it would be safe to run between Olivia and the motion detector would depend on how long it would take and how close to the motion detector you ran. If you ran less than 2 m from the motion detector, it would definitely be safe. f) If the motion detector was activated as soon as Olivia started to swing from at rest, the amplitude of each cycle of the graph would increase up until the amplitude became 6 m. Therefore, the graph would not be sinusoidal because the amplitude would be changing. 2. The radius of each wheel is equal to the ampli- tude of its respective graph, and the amplitude is half the distance between the maximum and minimum values. Since the minimum is - 1 and the maximum is 5 for paddle wheel A, its radius is 5 - (-1) • or 3 m. Since the minimum is - 1 and the maximum is 7 for paddle wheel B, its / - (-1) radius is 2 - or 4 m. The height of the axel of each wheel relative to the water is equal to the equation of the axis of its respective graph, and the equation of the axis is the equa- tion of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is - 1 and the maximum is 5 for paddle wheel A, the height of its axis relative to the water is y = - 5 + (-1) or y = 2 m. Since 2 the minimum is - 1 and the maximum is 7 for paddle wheel B, the height of its axis relative to the water is y = 7 + (-1) or y = 3 m. 2 The time taken for each wheel to complete one revolution is equal to the period of its respec- tive graph. and the period is the time it takes the graph to go through one complete cycle. Since the graph for paddle wheel A goes through one complete cycle between 0 s and 12 s, for example, the time it takes for it to com- plete one revolution is 12 - 0 or 12 s. Since the graph for paddle wheel B goes through one complete cycle between 0 s and 16 s, for exam- ple, the time it takes for it to complete one revolution is 16 - 0 or 16 s. The speed of each wheel is equal to its circumference divided by the time it takes for it to complete one revolu- tion. Since the circumference of paddle wheel A is m X 2 X 3 m or 6m m, and since the time it takes for it to complete one revolution is 12 s. 6т m its speed is • or 1.57 m/s. Since the circum- 12 s ference of paddle wheel B is + X 2 x 4 m or 8m m, and since the time it takes for it to complete one revolution is 16 s, its speed is 8т m - or 1.57 m/s. 16 s 3. This is one possible answer: a) The height above the cutting surface that the blade is set is equal to the maximum of the graph. Since the maximum is 1 in, the blade is set 1 in above the cutting surface. b) The period is the time it takes the function to go through one complete cycle. Since the function goes through one complete cycle between 0 s and 0.04 s, for example, the period is 0.04 - 0 or 0.04 s. In this case it represents how long it takes the blade to make a full rotation. c) The amplitude is half the distance between the maximum and minimum values. Since the minimum is -- 7 and the maximum is 1, the amplitude is or 4 in. In this case it 2 represents the radius of the saw blade. d) The speed of a tooth on the saw blade is equal to the saw blade's circumference divided by the time it takes for it to make a full rotation. Since the circumference of the saw blade is т X 2 X 4 in or 8m in, and since the time it takes for it to complete one revolution is 0.04 s, 8m in its speed is - 0.04 s or 628 in/s. a) The period is the time it takes the function to go through one complete cycle. Since the function goes through one complete cycle between 0 s and about 0.0325 s, for example. the period is about 0.0325 - 0 or 0.0325 s. b) The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the