632 CHAPTER 6 The Circular Functions and Their Graphs 56. Atmospheric Carbon Dioxide Refer to Exercise 55. The carbon dioxide content in the atmosphere at Barrow, Alaska, in parts per million (ppm) can be modeled by the function C1x2 = 0.04x2 + 0.6x + 330 + 7.5 sin 2px, where x = 0 corresponds to 1970. (Data from Zeilik, M., and S. Gregory, Introductory Astronomy and Astrophysics, Brooks/Cole.) (a) Graph C in the window 315, 454 by 3320, 4504. (b) What part of the function causes the amplitude of the oscillations in the graph of C to be larger than the amplitude of the oscillations in the graph of L in Exercise 55, which models Hawaii? 57. Average Daily Temperature The temperature in Anchorage, Alaska, can be approximated by the function T1x2 = 37 + 21 sin c 2p 365 1x - 912d , where T1x2 is the temperature in degrees Fahrenheit on day x, with x = 1 corresponding to January 1 and x = 365 corresponding to December 31. Use a calculator to estimate the temperature on each day. (Data from The World Almanac and Book of Facts.) (a) March 15 (day 74) (b) April 5 (day 95) (c) Day 200 (d) June 25 (e) October 1 (f ) December 31 58. Fluctuation in the Solar Constant The solar constant S is the amount of energy per unit area that reaches Earth’s atmosphere from the sun. It is equal to 1367 watts per m2 but varies slightly throughout the seasons. This fluctuation ΔS in S can be calculated using the formula ΔS = 0.034S sin c 2p182.5 - N2 365.25 d . In this formula, N is the day number covering a four-year period, where N = 1 corresponds to January 1 of a leap year and N = 1461 corresponds to December 31 of the fourth year. (Data from Winter, C., R. Sizmann, and L. L.Vant-Hull, Editors, Solar Power Plants, Springer-Verlag.) (a) Calculate ΔS for N = 80, which is the spring equinox in the first year. (b) Calculate ΔS for N = 1268, which is the summer solstice in the fourth year. (c) What is the maximum value of ΔS? (d) Find a value for N where ΔS is equal to 0. Musical Sound Waves Pure sounds produce single sine waves on an oscilloscope. Find the amplitude and period of each sine wave graph. On the vertical scale, each square represents 0.5. On the horizontal scale, each square represents 30° or p 6 . 59. 60. 61. Concept Check Compare the graphs of y = sin 2x and y = 2 sin x over the interval 30, 2p4. Can we say that, in general, sin bx = b sin x for b 70? Explain. 62. Concept Check Compare the graphs of y = cos 3x and y = 3 cos x over the interval 30, 2p4. Can we say that, in general, cos bx = b cos x for b 70? Explain.
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