666 CHAPTER 6 The Circular Functions and Their Graphs 20. Spring Motion A weight attached to a spring is pulled down 2 in. below the equilibrium position. (a) Assuming that the period is 1 3 sec, determine a model that gives the position of the weight at time t seconds. (b) What is the frequency? 21. Sound Wave When middle C is played on a piano, an oscilloscope produces a graph modeled by the function y = a sin vt having frequency 261.6 oscillations per sec. (a) What is the value of v? (b) What is the period of the function? 22. Sound Wave When the A key above middle C is played on a piano, an oscilloscope produces a graph modeled by the function y = a sin 880pt, where t is measured in seconds. Find the frequency of the sound wave. 23. Circadian Rhythm Body temperature is an example of a circadian rhythm — a 24-hour, internally controlled biological cycle. Many circadian rhythms can be modeled by trigonometric functions. Find a function T of the form T1t2 = a cos vt that models the increase and decrease in body temperature from average temperature, for a person whose temperature ranges from 97.7°F to 99.5°F. Let t = 0 correspond to 4 a.m., the time at which a person’s body temperature is lowest. 24. Circadian Rhythm Suppose that body temperature for a person whose temperature ranges from 97.9°F to 99.3°F is in circadian rhythm, where t = 0 corresponds to 4 p.m., the time at which the person’s body temperature is highest. Find a function T of the form T1t2 = a cos vt that models the increase and decrease in body temperature from average temperature. 25. Oscillating Cork When a rock is thrown into a pond, waves travel away from the rock across the surface of the pond. Suppose that a nearby cork is bobbing up and down in simple harmonic motion. If the height of the middle of the cork above water level after t seconds is given by s1t2 = 0.8 sin 18pt, where s(t) is measured in inches, how many times per second will the cork bob up and down? 26. Tides Suppose that during a 12-hour period, water height at the end of a pier begins at an average tide level of 13 ft, rises to a high tide level of 18 ft after 3 hr, falls to 8 ft at low tide 6 hr later, and returns to average tide level after another 3 hr. Find a function of the form h1t2 = a sin vt that models the increase and decrease in water height from average tide level after t hours.
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