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Study Guides > Precalculus II

Solutions for Modeling with Trigonometric Equations

Solutions to Try Its

1. The amplitude is  3\text{ }3, and the period is  23\text{ }\frac{2}{3}. 2.
x 3sin⁡(3x)3\sin \left(3x\right)
0 0
π6\frac{\pi }{6} 3
π3\frac{\pi }{3} 0
π2\frac{\pi }{2} −3-3
2π3\frac{2\pi }{3} 0
Graph of y=3sin(3x) using the five key points: intervals of equal length representing 1/4 of the period. Here, the points are at 0, pi/6, pi/3, pi/2, and 2pi/3. 3. y=8sin⁡(π12t)+32y=8\sin \left(\frac{\pi }{12}t\right)+32 The temperature reaches freezing at noon and at midnight. Graph of the function y=8sin(pi/12 t) + 32 for temperature. The midline is at 32. The times when the temperature is at 32 are midnight and noon. 4. initial displacement =6, damping constant = -6, frequency = 2π\frac{2}{\pi } 5. y=10e−0.5tcos⁡(πt)y=10{e}^{-0.5t}\cos \left(\pi t\right) 6. y=5cos⁡(6πt)y=5\cos \left(6\pi t\right)

Solutions to Odd-Numbered Exercises

1. Physical behavior should be periodic, or cyclical. 3. Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here. 5. y=−3cos⁡(π6x)−1y=-3\cos \left(\frac{\pi }{6}x\right)-1 7. 5sin⁡(2x)+25\sin \left(2x\right)+2 9. 4cos⁡(xπ2)−34\cos \left(\frac{x\pi }{2}\right)-3 11. 5−8sin⁡(xπ2)5 - 8\sin \left(\frac{x\pi }{2}\right) 13. tan⁡(xπ12)\tan \left(\frac{x\pi }{12}\right) 15. Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona. Graph of f(x) = -18cos(x*pi/12) - 5sin(x*pi/12) + 100 on the interval [0,24]. There is a single peak around 12. 17. 9 years from now 19. 56∘F56^\circ \text{F} 21. 1.80241.8024 hours 23. 4:30 25. From July 8 to October 23 27. From day 19 through day 40 29. Floods: July 24 through October 7. Droughts: February 4 through March 27 31. Amplitude: 11, period: 16\frac{1}{6}, frequency: 6 Hz 33. Amplitude: 5, period: 130\frac{1}{30}, frequency: 30 Hz 35. P(t)=−15cos⁡(π6t)+650+556tP\left(t\right)=-15\cos \left(\frac{\pi }{6}t\right)+650+\frac{55}{6}t 37. P(t)=−40cos⁡(π6t)+800(1.04)tP\left(t\right)=-40\cos \left(\frac{\pi }{6}t\right)+800{\left(1.04\right)}^{t} 39. D(t)=7(0.89)tcos⁡(40πt)D\left(t\right)=7{\left(0.89\right)}^{t}\cos \left(40\pi t\right) 41. D(t)=19(0.9265)tcos⁡(26πt)D\left(t\right)=19{\left(0.9265\right)}^{t}\cos \left(26\pi t\right) 43. 20.120.1 years 45. 17.8 seconds 47. Spring 2 comes to rest first after 8.0 seconds. 49. 500 miles, at 90∘{90}^{\circ } 51. y=6(5)x+4sin⁡(π2x)y=6{\left(5\right)}^{x}+4\sin \left(\frac{\pi }{2}x\right) 53. y=8(12)xcos⁡(π2x)+3y=8{\left(\frac{1}{2}\right)}^{x}\cos \left(\frac{\pi }{2}x\right)+3

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