SECTION 6.6 Phase Shift; Sinusoidal Curve Fitting 461 January, 1 February, 2 March, 3 April, 4 May, 5 June, 6 July, 7 August, 8 September, 9 October, 10 November, 11 December, 12 32.9 35.8 43.6 53.7 62.9 72.4 77.0 75.1 67.8 56.1 46.5 36.7 Source: U.S. National Oceanic & Atmospheric Administration Month, x Average Monthly Temperature, y (°F) 33. Tides The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Sunday, March 27, 2022, in Charleston, South Carolina, high tide occurred at 1:10 a.m. (1.2 hours) and low tide occurred at 7:44 a.m. (7.73 hours). Water heights are measured as the amounts above or below the mean lower low water.The height of the water at high tide was 5.73 feet, and the height of the water at low tide was −0.15 foot. (a) Approximately when did the next high tide occur? (b) Find a sinusoidal function of the form ω φ ( ) = − + y A x B sin that models the data. (c) Use the function found in part (b) to predict the height of the water at 5 pm. 34. Tides The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Friday, April 1, 2022, in Sitka,Alaska, high tide occurred at 3:43 a.m. (3.72 hours) and low tide occurred at 10:40 a.m. (10.67 hours). Water heights are measured as the amounts above or below the mean lower low water.The height of the water at high tide was 10.27 feet, and the height of the water at low tide was −1.80 feet. (a) Approximately when did the next high tide occur? (b) Find a sinusoidal function of the form ω φ ( ) = − + y A x B sin that models the data. (c) Use the function found in part (b) to predict the height of the water at 3 pm. 35. Hours of Daylight According to the Old Farmer’s Almanac, in Miami, Florida, the number of hours of sunlight on the summer solstice of 2022 was 13.75, and the number of hours of sunlight on the winter solstice was 10.52. (a) Find a sinusoidal function of the form ω φ ( ) = − + y A x B sin that models the data. (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year. (c) Draw a graph of the function found in part (a). (d) Look up the number of hours of sunlight for April 1 in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b). 36. Hours of Daylight According to the Old Farmer’s Almanac, in Detroit, Michigan, the number of hours of sunlight on the summer solstice of 2022 was 15.27, and the number of hours of sunlight on the winter solstice was 9.07. (a) Find a sinusoidal function of the form ω φ ( ) = − + y A x B sin that models the data. (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year. (c) Draw a graph of the function found in part (a). (d) Look up the number of hours of sunlight for April 1 in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b). 37. Hours of Daylight According to the Old Farmer’s Almanac, in Anchorage,Alaska, the number of hours of sunlight on the summer solstice of 2022 was 19.37, and the number of hours of sunlight on the winter solstice was 5.45. (a) Find a sinusoidal function of the form ω φ ( ) = − + y A x B sin that models the data. (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year. (c) Draw a graph of the function found in part (a). (d) Look up the number of hours of sunlight for April 1 in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b). 38. Hours of Daylight According to the Old Farmer’s Almanac, in Honolulu, Hawaii, the number of hours of sunlight on the summer solstice of 2022 was 13.42, and the number of hours of sunlight on the winter solstice was 10.83. (a) Find a sinusoidal function of the form ω φ ( ) = − + y A x B sin that models the data. (b) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year. (c) Draw a graph of the function found in part (a). (d) Look up the number of hours of sunlight for April 1 in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b). 39. Challenge Problem Coaster Motion A wooden roller coaster at Six Flags contains a run in the shape of a sinusoidal curve, with a series of hills. The crest of each hill is 106 feet above the ground. If it takes a car 1.8 seconds to go from the top of a hill to the bottom (4 feet off the ground), find a sinusoidal function of the form ω φ ( ) = − + y A t B sin that models the motion of the coaster train during this run starting at the top of a hill. Explaining Concepts 40. Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis. 41. Find an application in your major field that leads to a sinusoidal graph. Write an account of your findings.
RkJQdWJsaXNoZXIy NjM5ODQ=