Solutions to Try Its

1. The amplitude is $\text{ }3$, and the period is $\text{ }\frac{2}{3}$.

2.

x $3\sin \left(3x\right)$
0 0
$\frac{\pi }{6}$ 3
$\frac{\pi }{3}$ 0
$\frac{\pi }{2}$ $-3$
$\frac{2\pi }{3}$ 0

3. $y=8\sin \left(\frac{\pi }{12}t\right)+32$
The temperature reaches freezing at noon and at midnight.

4. initial displacement =6, damping constant = -6, frequency = $\frac{2}{\pi }$

5. $y=10{e}^{-0.5t}\cos \left(\pi t\right)$

6. $y=5\cos \left(6\pi t\right)$

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. $y=-3\cos \left(\frac{\pi }{6}x\right)-1$

7. $5\sin \left(2x\right)+2$

9. $4\cos \left(\frac{x\pi }{2}\right)-3$

11. $5 - 8\sin \left(\frac{x\pi }{2}\right)$

13. $\tan \left(\frac{x\pi }{12}\right)$

15. Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.

17. 9 years from now

19. $56^\circ \text{F}$

21. $1.8024$ 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: $\frac{1}{6}$, frequency: 6 Hz

33. Amplitude: 5, period: $\frac{1}{30}$, frequency: 30 Hz

35. $P\left(t\right)=-15\cos \left(\frac{\pi }{6}t\right)+650+\frac{55}{6}t$

37. $P\left(t\right)=-40\cos \left(\frac{\pi }{6}t\right)+800{\left(1.04\right)}^{t}$

39. $D\left(t\right)=7{\left(0.89\right)}^{t}\cos \left(40\pi t\right)$

41. $D\left(t\right)=19{\left(0.9265\right)}^{t}\cos \left(26\pi t\right)$

43. $20.1$ years

45. 17.8 seconds

47. Spring 2 comes to rest first after 8.0 seconds.

49. 500 miles, at ${90}^{\circ }$

51. $y=6{\left(5\right)}^{x}+4\sin \left(\frac{\pi }{2}x\right)$

53. $y=8{\left(\frac{1}{2}\right)}^{x}\cos \left(\frac{\pi }{2}x\right)+3$