6.5 Applications - Part I - Textbook 2 [Questions + Answers]

CHECK Your Understanding 1. The load on a trailer has shifted, causing the rear end of the trailer to swing left and right. The distance from onc of the tail lights on the trailer to the curb varies sinusoidally with time. The graph models this behaviour. a) What is the equation of the axis of the function, and what does it represent in this situation? b) What is the amplitude of the function, and what does it represent in this situation? • c) What is the period of the function, and what does it represent in this situation? d) Determine the equation and the range of the sinusoidal function. e) What are the domain and range of the function in terms of the situation? f) How far is the tail light from the curve at 1 = 3.2 s? 2. Don Quixote, a fictional character in a Spanish novel, attacked windmills because he thought they were giants. At one point, he got snagged by onc of the blades and was hoisted into the air. The graph shows his height above ground in terms of time. a) What is the equation of the axis of the function, and what does it represent in this situation? b) What is the amplitude of the function, and what does it represent in this situation? c) What is the period of the function, and what does it represent in this situation? d) If Don Quixote remains snagged for seven complete cycles, determine the domain and range of the function. e) Determine the equation of the sinusoidal function. f) If the wind speed decreased, how would that affect the graph of the sinusoidal function? 3. Chantelle is swinging back and forth on a trapeze. Her distance from a vertical support beam in terms of time can be modelled by a sinusoidal function. Ar 1 s, she is the maximum distance from the beam, 12 m. At 3 s, she is the minimum distance from the beam, 4 m. Determine an equation of a sinusoidal function that describes Chantelle's distance from the vertical beam in relation to time. PRACTISING 4. The interior and exterior, temperatures of an igloo were recorded over a 48 h • period. The data were collected and plotted, and two curves were drawn through the appropriate points. a) How are these curves similar? Explain how each of them might be related to this situation. b) Describe the domain and range of each curve. c) Assuming that the curves can be represented by sinusoidal functions, determine the equation of each function. 5. Skyscrapers sway in high-wind conditions. In one case, at t = 2 s, the top A floor of a building swayed 30 cm to the left (- 30 cm), and at 2 = 12, the cop floor swayed 30 cm to the right (+30 cm) of its starting position. a) What is the equation of a sinusoidal function that describes the motion of the building in terms of time? b) Dampers, in the forms of large tanks of water, are often added to the top floors of skyscrapers to reduce the severity of the sways. If a damper is added to this building, it will reduce the sway (not the period) by 70%. What is the equation of the new function that describes the motion of the building in terms of time? 6. Milton is floating in an inner tube in a wave pool. He rises 1.5 m from the bottom of the pool when he is at the trough of a wave. A stopwatch starts timing at this point. In 1.25 s, he is on the crest of the wave, 2.1 m from the bottom of the pool. a) Determine the equation of the function that expresses Milton's distance from the bottom of the pool in terms of time. b) What is the amplitude of the function, and what docs it represent in this situation? c) How far above the bottom of the pool is Milton at 1 = 4 s? d) If data are collected for only 40 s, how many complete cycles of the sinusoidal function will there be? e) If the period of the function changes to 3 s, what is the equation of this new function? 7. An oscilloscope hooked up to an alternating current (AC) circuit shows a sinc curve. The device records the current in amperes (A) on the vertical axis and the time in seconds on the horizontal axis. At t = 0 s, the current reads its maximum value of 4.5 A. At 7 = 120 s, the current reads its minimum value of -4.5 A. Determine the equation of the function that expresses the current in terms of time. 8. Candice is holding onto the end of a spring that is attached to a lead ball. As ' she moves her hand slightly up and down, the ball moves up and down. With a little concentration, she can repeatedly get the ball to reach a maximum height of 20 cm and a minimum height of 4 cm from the top of a surface. The first maximum height occurs at 0.2 s, and the first minimum height occurs at 0.6 s. a) Determine the equation of the sinusoidal function that represents the height of the lead ball in terms of time. b) Determine the domain and range of the function. c) What is the equation of the axis, and what does it represent in this situation? d) What is the height of the lead ball at 1.3 s? 9. A paintball is shot at a wheel of radius 40 cm. The paintball leaves a circular mark 10 cm from the outer edge of the wheel. As the wheel rolls, the mark moves in a circular motion. a) Assuming that the paintball mark starts at its lowest point, determine the equation of the sinusoidal function that describes the height of the mark in terms of the distance the wheel travels. b) If the wheel completes five revolutions before it stops, determine the domain and range of the sinusoidal function. c) What is the height of the mark when the wheel has travelled 120 cm from its initial position? 10. The population of rabbits, R(e), and the population of foxes, F(r), in a given region are modelled by the functions R(*) = 10 000 + 5000 cos (15t)° and F(:) = 1000 + 500 sin (15:)°, where r is the time in months. Referring to each graph, explain how the number of rabbits and the number of foxes are related. 11. What information would you need to determine an algebraic or graphical a model of a situation that could be modelled with a sinusoidal function? Extending 12. Two pulleys are connected by a belt. Pulley A has a radius of 3 cm, and Pulley B has a radius of 6 cm. As Pulley A rotates, a drop of paint on the circumference of Pulley B rotates around the axle of Pulley B. Initially, the paint drop is 7 cm above the ground. Determine the equation of a sinusoidal function that describes the height of the drop of paint above the ground in terms of the rotation of Pulley A. (a): Determine the equation of the function. (b): Evaluate (f(20)). (c): If (f(x)=2), then which of the following is true for x? (i): (180^{\circ}+360^{\circ}k, k\in I)   (ii): (360^{\circ}+180^{\circ}k. k\in I) (iii): (90^{\circ}+180^{\circ}k, k\in I) (iv): (270^{\circ}+360^{\circ}k, k\in I) (d): If (f(x)=-1), then which of the following is true for x? (i): (180^{\circ}+360^{\circ}k, k\in I) (ii): (360^{\circ}+90^{\circ}k, k\in I) (iii): (90^{\circ}+360^{\circ}k, k\in I) (iv): (90^{\circ}+180^{\circ}k. k\in I) Question 14: Using graphing technology, determine x when (f(x)=7) for the function (f(x)=4~cos(2x)^{\circ}+3) in the domain ({x\in R|0^{\circ}\le x\le360^{\circ}}). 6.7 Solving Problems Using Sinusoidal Models, pp. 398-401 1. Distance Between the Tail Light and the Curb f(t) Distance (m) 1.5 1.0 0.5 Time (s) a) The equation of the axis is the equation of the horizontal line that is halfway berween the max- imum and the minimum. Since the minimum is 1.0 and the maximum is 2.0, the equation of the axis is d = 2.0 + 1.0 - or d = 1.5 m. In this situa- tion it represents the distance between the tail lights and the curb if the trailer isn't swinging back and forth. b) The amplitude is half the distance between the maximum and minimum values. Since the minimum is 1.0 and the maximum is 2.0, the 2.0 - 1.0 amplitude is or 0.5 m. In this situation 2 it represents the distance the trailer swings to the left and right. c) The period is the time it takes the function to go through one complete cycle. Since the func- tion goes through one complete cycle between 0 s and 2 s, for example, the period is 2 - 0 or 2 s. In this situation it represents the time it takes the trailer to swing back and forth. d) Since the amplitude is 0.5 m. a in the function y = a cos (k(x - d)) + c is 0.5. Since the period is 2 s, k in the function 360 y = a cos (k(x - d)) + c is - - or 180. Since 2 the equation of the axis is y = 1.5 m, c in the function y = a cos(k(x - d)) + c is 1.5. Since the function would have crossed the y-axis at its maximum had it not been shifted 0.5 units to the right, d in the equation y = a cos (k(x - d)) is 0.5. Therefore, the equation is d = 0.5 cos 180(t - 0.5)J° + 1.5. The range is all possible values of y. Since y oscillates between 1 and 2, the range is {d € R11 ≤ d ≤ 2}. e) The range is the distance the trailer swings back and forth. The domain is the time. 1) Since the equation that represents the sifuation is d = 0.5 cos [180 (z - 0.5)J° + 1.5, the dis- tance from the tail light to the curve at 1 = 3.2 s is d = 0.5 cos [180 (3.2 - 0.5)J° + 1.5 = 0.5 cos 486° ÷ 1.5 = 0.5 X -0.59 ÷ 1.5 = 1.2 ml time (s) a) The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the mini- mum is 3 and the maximum is 17, the equation of the axis is h = 17 + 3 or h = 10 m. In this situation it represents the axle height. b) The amplitude is half the distance between the maximum and minimum values. Since the minimum is 3 and the maximum is 17, the amplitude is 17 - 3 or 7 m. In this situation it represents the length of the blade. c) The period is the time it takes the function to go through one complete cycle. Since the func- tion goes through one complete cycle between 0 s and 20 s, for example, the period is 20 - 0 or 20 s. In this situation it represents the time to compiete one revolution. d) The domain is all possible values of r. Since Don Quixote remains snagged for 7 complete cycles and each cycle takes 20 s, the domain is (1 € R10 ≤ 1 ≤ 140}. The range is all possible values of h. Since h oscillates between 3 and 17, the range is (h € R13 ≤ h ≤ 17}. e) Since the amplitude is 7 m, a in the func- tion y = a cos(k(x - d)) + c is 7. Since the period is 20 s, k in the function 360 y = a cos(k(x - d)) + c is 20 or 18. Since the equation of the axis is h = 10 m, c in the function y = a cos(k(x - d)) + c is 10. Since the function crosses the y-axis at its minimum and is reflected in its axis, the sign of a in the equa- tion y = a cos (k(x - d)) is negative. Therefore, the equation is h = - 7 cos (18x)° + 10. f) If the wind speed decreased, the period would be larger. 3. The amplitude is half the distance between the maximum and minimum values. Since the minimum is 4 and the maximum is 12, the 12 - 4 amplitude is • or 4. Since the amplitude 2 is 4, a in the equation y = a cos (k(x - d)) + c is 4. The period is the change in x that occurs as the function goes through one complete cycle. Since Chantelle goes through one complete cycle between x = 1 s and x = 5 s, for example, the period is 5 - 1 or 4 s. Since the period is 4 s, k in the equation y = a cos (k(x - d)) + c 360 is or 90. The equation of the axis is the 4 equation of the horizontal line that is halfway between the maximum and the minimum. Since the minimum is 4 and the maximum is 12, the 12 + 4 equation of the axis is y = - or y = 8. Since the equation of the axis is y = 8, c in the equation y = a cos (k(x - d)) + c is 8. Since the function would have crossed the y-axis at its maximum value had it not been shifted 1 unit to the right, the equation for it should use the cosine function and be shifted 1 unit to the right. Therefore, d in the equation y = a cos (k(x - d)) is 1. Therefore, the equation for the function is d = 4 cos (90(1 - 1))º + 8. a) They have the same period (24) and the same horizontal translation (12), but different ampli- tudes (2.5 and 10) and different equations of the axis (7 = 17.5 and 7 = - 20). The top curve is probably the interior temperature (higher, with less fluctuation), while the bottom curve is probably the exterior temperature. b) The domain is all possible values of t. Since both curves cover a 48 h period, the domain for both curves is {7 € R10 $ 1 ≤ 48}. The range is all possible values of T. Since T oscillates between 15 and 20 for the top curve, the range for the top curve is {7 € R115 ≤ T ≤ 20}. Since T oscillates between - 30 and - 10 for the bottom curve, the range for the bottom curve is {TERI -30 ST≤ - 10}. c) i) Since the amplitude for the top curve is 2.5, a in the function y = a cos (k(x - d)) + c is 2.5. Since the period is 24 h, k in the func- tion y = a cos(k(x - d)) + c is 360 • or 15. Since the equation of the axis is y = 17.5, c in the function y = a cos(k(x - d)) + c is 17.5. Since the function would have crossed the y-axis at its maximum had it not been shifted 12 units to the right, d in the equation y = a cos (k(x - d)) is 12. Therefore, the equation for the top curve is T = 2.5 cos(15 (h - 12)J° + 17.5. ii) Since the amplitude for the bottom curve is 10, a in the function y = a cos (k(x - d)) + c is 10. Since the period is 24 h, k in the func- 360 tion y = a cos (k(x - d)) + c is or 15. 24 Since the equation of the axis is y = - 20, c in the function y = a cos(k(x - d)) + c is -20. Since the function would have crossed the y-axis at its maximum had it not been shifted 12 units to the right. d in the equation y = a cos(k(x - d)) is 12. Therefore, the equation for the top curve is T = 10 cos 15(h - 12)}° - 20. 5. a) The amplitude is half the distance between the maximum and minimum values. Since the minimum is -- 30 and the maximum is 30, 30 - (- 30) the amplitude is - or 30. Since the amplitude is 30, a in the equation y = a cos(k(x - d)) + c is 30. The period is the change in x that occurs as the function goes through one complete cycle. Since the building goes through one complete cycle between 1 = 2 s and 1 = 22 s, for example, the period is 22 - 2 or 20 s. Since the period is 20 s, k in the 360 equation y = a cos (k (x - d)) + c is - or 18. 20 The equation of the axis is the equation of the horizontal line that is halfway between the max- imum and the minimum. Since the minimum is -30 and the maximum is 30. the equation of value had it not been shifted 12 units to the right. the equation for it should use the cosine function and be shifted 12 units to the right. Therefore. d in the equation y = a cos(k(x - d)) is 12. Therefore, the equation for the function is d = 30 cos (18(1 - 12))°. b) If the sway is reduced by 70 percent, the amplitude of the function is reduced by 70 percent. Everything else about the function remains the same. Therefore, the function becomes d = 30 x 0.3 cos (18(1 - 12))° or d = 9 cos (18(2 - 12))°. 6. a) The amplitude is half the distance between the maximum and minimum values. Since the minimum is 1.5 m and the maximum is 2.1 m, the amplitude is = 2.1 - 1.5 - ог 0.3 m. 2 Since the amplitude is 0.3 m, a in the equation y = a cos(k(x - d)) + c is 0.3. The period is the change in 7 that occurs as the function goes through one complete cycle. Since Milton goes through one complete cycle between 1 = 0 s and 1 = 2.5 s, for example, the period is 2.5 - 0 or 2.5 s. Since the period is 2.5 s, k in the 360 equation y = a cos(k(x - d)) + c is 2.5 144. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the mini- mum is 1.5 m and the maximum is 2.1 m, the equation of the axis is y = 21 + 15 • от y = 1.8. Since the equation of the axis is y = 1.8, c in the equation y = a cos (k(x - d)) + c is 1.8. Since the function crosses the y-axis at its minimum value, the equation for it should use the cosine function and be reflected in its axis. Therefore, the sign of a in the equation y = a cos(k(x - d)) is negative. Therefore, the equation for the function is d = - 0.3 cos (144г)° + 1.8. b) The amplitude is half the distance between the maximum and minimum values. Since the minimum is 1.5 m and the maximum is 2.1 m. 2.1 - 1.5 the amplitude is - or 0.3 m. In this 2 situation it represents the height of the crest relative to the normal water level. c) Since the equation for this situation is d = -0.3 cos (1441)° + 1.8, at / = 4 s, Milton's height above the bottom of the pool is d = -0.3 cos (144 × 4)° + 1.8 - - 0.3 cos (576)° + 1.8 = - 0.3 × -0.8 + 1.8 = 0.2 + 1.8 = 2 m d) If data are collected for only 40 s, there will be 40 ÷ 2.5 or 16 complete cycles. e) If the period of the function changes to 3 s. k in the equation y = a cos (k(x - d)) + c is 360 3 - or 120. Therefore, the equation becomes d = - 0.3 cos (120r)° + 1.8. 7. The amplitude is half the distance between the maximum and minimum values. Since the minimum is - 4.5 A and the maximum is 4.5 A, the amplitude is - 45 - (-4.5) от 4.5 A. Since the amplitude is 4.5 A, a in the equation y = a cos (k(x - d)) + c is 4.5. The period is the change in 1 that occurs as the function goes through one complete cycle. Since the function goes through one complete cycle between 1 = 0 s and 1 = 1 60 s, for example, the period is - 0 60 or s. Since the period is s, k in the 60 60 equation y = a cos(k(x - d)) + c is 360 or 60 21 600. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the min- imum is -4.5 A and the maximum is 4.5 A, the equation of the axis is y = 45 + (-4.5) y = 0. Since the equation of the axis is y = 0, c in the equation y = a cos(k(x - d)) + c is 0. Since the function crosses the y-axis at its maxi- mum value, the equation for it should use the cosine function. Therefore, the equation for the function is C = 4.5 cos (21 600r)°. 8. a) The amplitude is half the distance between the maximum and minimum values. Since the minimum is 4 cm and the maximum is 20 cm. 20 - 4 the amplitude is or 8 cm. Since the amplitude is 8 cm. a in the equation y = a cos (k(x - d)) + c is 8. The period is the change in t that occurs as the function goes through one complete cycle. Since the function goes through one complete cycle between 1 = 0.2 s and 7 = 1 s, for example, the period is 1 - 0.2 or 0.8 s. Since the period is 0.8 s, k in the equation y = a cos(k(x - d)) + c is - 360 0.8 or 450. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the min- imum is 4 cm and the maximum is 20 cm, the 20 + 4 equation of the axis is y = = 2 - от у = 12. Since the equation of the axis is y = 12, c in the equation y = a cos (k(x - d)) + c is 12. Since the function would have crossed the y-axis at its maximum value had it not been shifted 0.2 units to the right, the equation for it should use the cosine function and be shifted 0.2 units to the right. Therefore, d in the equation y = a cos (k(x - d)) is 0.2. Therefore, the equation for the function is h = 8 cos (450(t - 0.2))° + 12. b) The domain is all possible values of 1. Since there is no limit to the amount of time that Candice can hold onto the spring with the lead ball, the domain is {t € R}. The range is all possible values of h. Since the height of the ball oscillates between 4 cm and 20 cm, the range is {h € R14 sh ≤ 20}. c) The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the mini- mum is 4 cm and the maximum is 20 cm, the 20 + 4 equation of the axis is h = - - or h = 12 cm. 2 In this situation it represents the resting position of the spring. d) Since the equation for the situation is h = 8 cos (450(1 - 0.2))° + 12, the height of the lead ball at 1 = 1.3 s is h = 8 cos (450(1.3 - 0.2))° + 12 = 8 cos (450 X 1.1)° + 12 = 8 cos (495)° + 12 = 8 x - 0.71 + 12 = -5.7 + 12 = 6.3 cm 9. a) The amplitude is half the distance between the maximum and minimum values. Since the minimum is 10 and the maximum is 70, the 70 - 10 amplitude is - - or 30. Since the amplitude 2 is 30, a in the equation y = a cos (k(x - d)) + c is 30. The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = 0 cm and x = 2 X + X 40 ог 251.3 cm, the period is 251.3 - 0 or 251.3 cm. Since the period is 251.3 cm, k in the 360 equation y = a cos(k(x - d)) + c is 251.3 1.43. The equation of the axis is the equation of the horizontal line that is halfway between the maximum and the minimum. Since the mini- mum is 10 and the maximum is 70, the equation of the axis is y = 70 + 10 - or y = 40. Since the equation of the axis is y = 40, c in the equation y = a cos(k(x - d)) + c is 40. Since the func- tion crosses the y-axis at its minimum value, the equation for it should use the cosine function and be reflected in its axis. Therefore, the sign of a in the equation y = a cos (k(x - d)) is negative. Therefore, the equation for the func- tion is h = - 30 cos (1.43x)° + 40. b) The domain is all possible values of d. Since the wheel completes 5 revolutions before it stops, it travels 2 X # X 40 × 5 or 400m cm before it stops. Therefore, the domain is {d E R10 ≤ d ≤ 400}. The range is all possi- ble values of h. Since the height of the circular mark oscillates between 10 cm and 70 cm, the range is (h € R 110 ≤ h ≤ 70}. c) Since the equation for the situation is h = - 30 cos (1.43x)° + 40, the height of the mark when the wheel has traveled 120 cm is / = - 30 cos (1.43 × 120)* + 40 = - 30 cos (171.6)° ÷ 40 = - 30 × - 0.99 ÷ 40 = 60 7 cm 10. The periods of the two functions are the same, so the population of rabbits and the popu- lation of foxes both go through one complete cycle in the same amount of time. That's the only way the population of rabbits and the pop- ulation of foxes appear to be related. The ampli- tudes, equations of the axis, and horizontal translations of the two functions all differ. 11. The period and amplitude, as well as where it starts on the x-axis and the position on the y-axis when it started. 12. The amplitude is half the distance between the maximum and minimum values. Since the minimum is 7 cm and the maximum is 19 cm. 19 - 7 the amplitude is or 6 cm. Since the amplitude is 6 cm, a in the equation y = a cos(k(x - d)) + c is 6. The period is the change in x that occurs as the function goes through one complete cycle. Since pulley A has half the circumference as pulley B, pulley A makes 2 rotations for every 1 rotation of pulley B. Therefore, the change in x in terms of pulley A that occurs as the function goes through one complete cycle is 2 X 360° or 720°. Since the graph goes through one complete cycle between x = 0 and x = 720, the period is 720 - 0 or 720. Since the period is 720, k in the equation 360 y = a cos(k(x - d)) + c is - 720 or 0.5. The equation of the axis is the equation of the hori- zontal line that is halfway between the maxi- mum and the minimum. Since the minimum is 7 cm and the maximum is 19 cm, the equation of the axis is y = 19 + 7 - or y = 13. Since the equation of the axis is y = 13, c in the equation y = a cos (k(x - d)) + c is 13. Since the func- tion crosses the y-axis at its minimum value, the equation for it should use the cosine function and be reflected in its axis. Therefore, the sign of a in the equation y = a cos (k(x - d)) is negative. Therefore, the equation for the func- tion is h = - 6 cos (0.5x)° + 13. 13. a) The amplitude is half the distance between the maximum and minimum values. Since the minimum is - 4 and the maximum is 2, the amplitude is 2 - (-4) • or 3. Since the amplitude is 3, a in the equation y = a cos(k(x - d)) + c is 3. The period is the change in x that occurs as the function goes through one complete cycle. Since the graph goes through one complete cycle between x = 0 and x = 360, the period is 360 - 0 or 360. Since the period is 360, k in the equation y = a cos(k(x - d)) + c is 360 or 1. The equation of the axis is the equation of the horizontal line that is halfway berween the maximum and the minimum. Since the minimum is - 4 and the maximum is 2, the equation of the axis is y = 2 + (-4) or y = - 1.