{"id":1403,"date":"2023-06-05T14:51:23","date_gmt":"2023-06-05T14:51:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-modeling-with-trigonometric-equations\/"},"modified":"2023-06-05T14:51:23","modified_gmt":"2023-06-05T14:51:23","slug":"solutions-for-modeling-with-trigonometric-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-modeling-with-trigonometric-equations\/","title":{"raw":"Solutions 56: Modeling with Trigonometric Equations","rendered":"Solutions 56: Modeling with Trigonometric Equations"},"content":{"raw":"\nSolutions to Odd-Numbered Exercises<\/h2>\n1. Physical behavior should be periodic, or cyclical.\n\n3. Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.\n\n5. [latex]y=-3\\cos \\left(\\frac{\\pi }{6}x\\right)-1[\/latex]\n\n7. [latex]5\\sin \\left(2x\\right)+2[\/latex]\n\n9. [latex]4\\cos \\left(\\frac{x\\pi }{2}\\right)-3[\/latex]\n\n11. [latex]5 - 8\\sin \\left(\\frac{x\\pi }{2}\\right)[\/latex]\n\n13. [latex]\\tan \\left(\\frac{x\\pi }{12}\\right)[\/latex]\n\n15. Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.\n\n![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27164217\/CNX_Precalc_Figure_07_06_202.jpg\")
\n\n17. 9 years from now\n\n19. [latex]56^\\circ \\text{F}[\/latex]\n\n21. [latex]1.8024[\/latex] hours\n\n23. 4:30\n\n25. From July 8 to October 23\n\n27. From day 19 through day 40\n\n29. Floods: July 24 through October 7. Droughts: February 4 through March 27\n\n31. Amplitude: 11, period: [latex]\\frac{1}{6}[\/latex], frequency: 6 Hz\n\n33. Amplitude: 5, period: [latex]\\frac{1}{30}[\/latex], frequency: 30 Hz\n\n35. [latex]P\\left(t\\right)=-15\\cos \\left(\\frac{\\pi }{6}t\\right)+650+\\frac{55}{6}t[\/latex]\n\n37. [latex]P\\left(t\\right)=-40\\cos \\left(\\frac{\\pi }{6}t\\right)+800{\\left(1.04\\right)}^{t}[\/latex]\n\n39. [latex]D\\left(t\\right)=7{\\left(0.89\\right)}^{t}\\cos \\left(40\\pi t\\right)[\/latex]\n\n41. [latex]D\\left(t\\right)=19{\\left(0.9265\\right)}^{t}\\cos \\left(26\\pi t\\right)[\/latex]\n\n43. [latex]20.1[\/latex] years\n\n45. 17.8 seconds\n\n47. Spring 2 comes to rest first after 8.0 seconds.\n\n49. 500 miles, at [latex]{90}^{\\circ }[\/latex]\n\n51. [latex]y=6{\\left(5\\right)}^{x}+4\\sin \\left(\\frac{\\pi }{2}x\\right)[\/latex]\n\n53. [latex]y=8{\\left(\\frac{1}{2}\\right)}^{x}\\cos \\left(\\frac{\\pi }{2}x\\right)+3[\/latex]\n","rendered":"Solutions to Odd-Numbered Exercises<\/h2>\n
1. Physical behavior should be periodic, or cyclical.<\/p>\n
3. Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.<\/p>\n
5. [latex]y=-3\\cos \\left(\\frac{\\pi }{6}x\\right)-1[\/latex]<\/p>\n
7. [latex]5\\sin \\left(2x\\right)+2[\/latex]<\/p>\n
9. [latex]4\\cos \\left(\\frac{x\\pi }{2}\\right)-3[\/latex]<\/p>\n
11. [latex]5 - 8\\sin \\left(\\frac{x\\pi }{2}\\right)[\/latex]<\/p>\n
13. [latex]\\tan \\left(\\frac{x\\pi }{12}\\right)[\/latex]<\/p>\n
15. Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.<\/p>\n
<\/p>\n
17. 9 years from now<\/p>\n
19. [latex]56^\\circ \\text{F}[\/latex]<\/p>\n
21. [latex]1.8024[\/latex] hours<\/p>\n
23. 4:30<\/p>\n
25. From July 8 to October 23<\/p>\n
27. From day 19 through day 40<\/p>\n
29. Floods: July 24 through October 7. Droughts: February 4 through March 27<\/p>\n
31. Amplitude: 11, period: [latex]\\frac{1}{6}[\/latex], frequency: 6 Hz<\/p>\n
33. Amplitude: 5, period: [latex]\\frac{1}{30}[\/latex], frequency: 30 Hz<\/p>\n
35. [latex]P\\left(t\\right)=-15\\cos \\left(\\frac{\\pi }{6}t\\right)+650+\\frac{55}{6}t[\/latex]<\/p>\n
37. [latex]P\\left(t\\right)=-40\\cos \\left(\\frac{\\pi }{6}t\\right)+800{\\left(1.04\\right)}^{t}[\/latex]<\/p>\n
39. [latex]D\\left(t\\right)=7{\\left(0.89\\right)}^{t}\\cos \\left(40\\pi t\\right)[\/latex]<\/p>\n
41. [latex]D\\left(t\\right)=19{\\left(0.9265\\right)}^{t}\\cos \\left(26\\pi t\\right)[\/latex]<\/p>\n
43. [latex]20.1[\/latex] years<\/p>\n
45. 17.8 seconds<\/p>\n
47. Spring 2 comes to rest first after 8.0 seconds.<\/p>\n
49. 500 miles, at [latex]{90}^{\\circ }[\/latex]<\/p>\n
51. [latex]y=6{\\left(5\\right)}^{x}+4\\sin \\left(\\frac{\\pi }{2}x\\right)[\/latex]<\/p>\n
53. [latex]y=8{\\left(\\frac{1}{2}\\right)}^{x}\\cos \\left(\\frac{\\pi }{2}x\\right)+3[\/latex]<\/p>\n\n\t\t\t