709 7.3 Sum and Difference Identities (Modeling) Solve each problem. See Example 6. 103. Electric Current The voltage V in a typical 115-volt outlet can be expressed by the function V1t2 = 163 sin 120pt, where 120p is the angular speed (in radians per second) of the rotating generator at an electrical power plant, and t is time in seconds. (Data from Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall.) (a) How many times does the current oscillate in 0.05 sec? (b) What are the maximum and minimum voltages in this outlet? (c) Is the voltage always equal to 115 volts? 104. Sound Waves Sound is a result of waves applying pressure to a person’s eardrum. For a pure sound wave radiating outward in a spherical shape, the trigonometric function P = a r cos a 2pr l - ctb can be used to model the sound pressure at a radius of r feet from the source, where t is time in seconds, l is length of the sound wave in feet, c is speed of sound in feet per second, and a is maximum sound pressure at the source measured in pounds per square foot. (Data from Beranek, L., Noise and Vibration Control, Institute of Noise Control Engineering, Washington, D.C.) Let l = 4.9 ft and c = 1026 ft per sec. (a) Let a = 0.4 lb per ft2. Graph the sound pressure at distance r = 10 ft from its source in the window 30, 0.054 by 3-0.05, 0.054. Describe P at this distance. (b) Now let a = 3 and t = 10. Graph the sound pressure in the window 30, 204 by 3-2, 24. What happens to pressure P as radius r increases? (c) Suppose a person stands at a radius r so that r = nl, where n is a positive integer. Use the difference identity for cosine to simplify P in this situation. 105. Back Stress If a person bends at the waist with a straight back making an angle of u degrees with the horizontal, then the force F exerted on the back muscles can be modeled by the equation F = 0.6W sin1u + 90°2 sin 12° , where W is the weight of the person. (Data from Metcalf, H., Topics in Classical Biophysics, Prentice-Hall.) (a) Calculate force F, to the nearest pound, for W= 170 lb and u = 30°. (b) Use an identity to show that F is approximately equal to 2.9W cos u. (c) For what value of u is F maximum? 106. Back Stress Refer to Exercise 105. (a) Suppose a 200-lb person bends at the waist so that u = 45°. Calculate the force, to the nearest pound, exerted on the person’s back muscles. (b) Approximate graphically the value of u, to the nearest tenth, that results in the back muscles of a 200-lb person exerting a force of 400 lb.
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