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. [latex]y=-3\cos \left(\frac{\pi }{6}x\right)-1[/latex]
7. [latex]5\sin \left(2x\right)+2[/latex]
9. [latex]4\cos \left(\frac{x\pi }{2}\right)-3[/latex]
11. [latex]5 - 8\sin \left(\frac{x\pi }{2}\right)[/latex]
13. [latex]\tan \left(\frac{x\pi }{12}\right)[/latex]
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.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27164217/CNX_Precalc_Figure_07_06_202.jpg)
17. 9 years from now
19. [latex]56^\circ \text{F}[/latex]
21. [latex]1.8024[/latex] 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: [latex]\frac{1}{6}[/latex], frequency: 6 Hz
33. Amplitude: 5, period: [latex]\frac{1}{30}[/latex], frequency: 30 Hz
35. [latex]P\left(t\right)=-15\cos \left(\frac{\pi }{6}t\right)+650+\frac{55}{6}t[/latex]
37. [latex]P\left(t\right)=-40\cos \left(\frac{\pi }{6}t\right)+800{\left(1.04\right)}^{t}[/latex]
39. [latex]D\left(t\right)=7{\left(0.89\right)}^{t}\cos \left(40\pi t\right)[/latex]
41. [latex]D\left(t\right)=19{\left(0.9265\right)}^{t}\cos \left(26\pi t\right)[/latex]
43. [latex]20.1[/latex] years
45. 17.8 seconds
47. Spring 2 comes to rest first after 8.0 seconds.
49. 500 miles, at [latex]{90}^{\circ }[/latex]
51. [latex]y=6{\left(5\right)}^{x}+4\sin \left(\frac{\pi }{2}x\right)[/latex]
53. [latex]y=8{\left(\frac{1}{2}\right)}^{x}\cos \left(\frac{\pi }{2}x\right)+3[/latex]
