752 CHAPTER 7 Trigonometric Identities and Equations 111. Hearing Difference Tones When a musical instrument creates a tone of 110 Hz, it also creates tones at 220, 330, 440, 550, 660, . . . Hz. A small speaker cannot reproduce the 110-Hz vibration but it can reproduce the higher frequencies, which are the upper harmonics. The low tones can still be heard because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is 110 Hz, and this difference tone will be heard by a listener. (Data from Benade, A., Fundamentals of Musical Acoustics, Dover Publications.) (a) In the window 30, 0.034 by 3-1, 14, graph the upper harmonics represented by the pressure P = 1 2 sin32p12202t4 + 1 3 sin32p13302t4 + 1 4 sin32p14402t4. (b) Estimate all t-coordinates where P is maximum. (c) What does a person hear in addition to the frequencies of 220, 330, and 440 Hz? (d) Graph the pressure produced by a speaker that can vibrate at 110 Hz and above. 112. Daylight Hours The seasonal variation in length of daylight can be modeled by a sine function. For example, the daily number of hours of daylight in New Orleans is given by the equation h = 35 3 + 7 3 sin 2px 365 , where x is the number of days after March 21 (disregarding leap year). (Data from Bushaw, D., et al., A Sourcebook of Applications of School Mathematics, National Council of Teachers of Mathematics.) (a) On what date will there be about 14 hr of daylight? (b) What date has the least number of hours of daylight? (c) When will there be about 10 hr of daylight? 113. Average Monthly Temperature The following function approximates average monthly temperature y (in °F) in Vancouver, Canada. Here x represents the month, where x = 1 corresponds to January, x = 2 corresponds to February, and so on. (Data from www.weatherbase.com) ƒ1x2 = 13 sin c p 6 1x - 42d + 51 When is the average monthly temperature (a) 64°F (b) 39°F? 114. Average Monthly Temperature The following function approximates average monthly temperature y (in °F) in Phoenix, Arizona. Here x represents the month, where x = 1 corresponds to January, x = 2 corresponds to February, and so on. (Data from National Climatic Data Center.) ƒ1x2 = 20 cos c p 6 1x - 72d + 75 When is the average monthly temperature (a) 75°F (b) 60°F? (Modeling) Alternating Electric Current The study of alternating electric current requires solving equations of the form i = Imax sin 2pƒt, for time t in seconds, where i is instantaneous current in amperes, Imax is maximum current in amperes, and f is the number of cycles per second. (Data from Hannon, R. H., Basic Technical Mathematics with Calculus, W. B. Saunders Company.) Find the least positive value of t, given the following data. 115. i = 40, Imax = 100, f = 60 116. i = 50, Imax = 100, f = 120 117. i = Imax , f = 60 118. i = 1 2 Imax , f = 60
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