Trigonometric Functions Name:

Modeling with Trigono-metric Functions

Date:

1. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 20 feetoccurred at 6:14 and a low tide of 4 feet occurred at 12:28.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 13:29?c) How long (in hours) is the tide (height of the water) 13 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 8 cos

(30π (t− 187

30 )187

)+ 12

b) h(13 : 29) = 5.03 feetc) ∆t = 5.74 hoursd) hmin = 4 feet, hmax = 20 feet

2. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 22 feetoccurred at 7:25 and a low tide of 6 feet occurred at 13:39.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 13:29?c) How long (in hours) is the tide (height of the water) 14 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 8 cos

(30π (t− 89

12)187

)+ 14

b) h(13 : 29) = 6.03 feetc) ∆t = 6.23 hoursd) hmin = 6 feet, hmax = 22 feet

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3. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 24 feetoccurred at 4:7 and a low tide of 6 feet occurred at 10:19.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 5:23?c) How long (in hours) is the tide (height of the water) 18 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(5π (t− 247

60 )31

)+ 15

b) h(5 : 23) = 22.21 feetc) ∆t = 4.86 hoursd) hmin = 6 feet, hmax = 24 feet

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4. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 22 feetoccurred at 2:16 and a low tide of 6 feet occurred at 8:26.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 13:22?c) How long (in hours) is the tide (height of the water) 15 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 8 cos

(6π (t− 34

15)37

)+ 14

b) h(13 : 22) = 20.47 feetc) ∆t = 5.67 hoursd) hmin = 6 feet, hmax = 22 feet

5. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 22 feetoccurred at 2:6 and a low tide of 4 feet occurred at 8:18.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 7:27?c) How long (in hours) is the tide (height of the water) 13 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(5π (t− 21

10)31

)+ 13

b) h(7 : 27) = 4.82 feetc) ∆t = 6.2 hoursd) hmin = 4 feet, hmax = 22 feet

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6. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 21 feetoccurred at 6:5 and a low tide of 3 feet occurred at 12:15.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 6:25?c) How long (in hours) is the tide (height of the water) 14 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(6π (t− 73

12)37

)+ 12

b) h(6 : 25) = 20.87 feetc) ∆t = 5.29 hoursd) hmin = 3 feet, hmax = 21 feet

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7. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 23 feetoccurred at 2:28 and a low tide of 7 feet occurred at 8:42.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 8:22?c) How long (in hours) is the tide (height of the water) 17 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 8 cos

(30π (t− 37

15)187

)+ 15

b) h(8 : 22) = 7.11 feetc) ∆t = 5.23 hoursd) hmin = 7 feet, hmax = 23 feet

8. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 23 feetoccurred at 11:4 and a low tide of 1 feet occurred at 17:18.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 20:26?c) How long (in hours) is the tide (height of the water) 16 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 11 cos

(30π (t− 166

15 )187

)+ 12

b) h(20 : 26) = 12.09 feetc) ∆t = 4.76 hoursd) hmin = 1 feet, hmax = 23 feet

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9. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 21 feetoccurred at 6:22 and a low tide of 3 feet occurred at 12:32.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 15:25?c) How long (in hours) is the tide (height of the water) 14 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(6π (t− 191

30 )37

)+ 12

b) h(15 : 25) = 11.08 feetc) ∆t = 5.29 hoursd) hmin = 3 feet, hmax = 21 feet

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10. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 22 feetoccurred at 7:21 and a low tide of 4 feet occurred at 13:35.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 15:28?c) How long (in hours) is the tide (height of the water) 13 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(30π (t− 147

20 )187

)+ 13

b) h(15 : 28) = 7.76 feetc) ∆t = 6.23 hoursd) hmin = 4 feet, hmax = 22 feet

11. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 24 feetoccurred at 3:19 and a low tide of 4 feet occurred at 9:31.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 12:22?c) How long (in hours) is the tide (height of the water) 17 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 10 cos

(5π (t− 199

60 )31

)+ 14

b) h(12 : 22) = 12.74 feetc) ∆t = 5.0 hoursd) hmin = 4 feet, hmax = 24 feet

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12. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 22 feetoccurred at 9:29 and a low tide of 2 feet occurred at 15:43.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 12:27?c) How long (in hours) is the tide (height of the water) 13 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 10 cos

(30π (t− 569

60 )187

)+ 12

b) h(12 : 27) = 12.76 feetc) ∆t = 5.84 hoursd) hmin = 2 feet, hmax = 22 feet

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13. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 23 feetoccurred at 11:10 and a low tide of 5 feet occurred at 17:20.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 12:28?c) How long (in hours) is the tide (height of the water) 17 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(6π (t− 67

6 )37

)+ 14

b) h(12 : 28) = 21.1 feetc) ∆t = 4.83 hoursd) hmin = 5 feet, hmax = 23 feet

14. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 23 feetoccurred at 0:11 and a low tide of 5 feet occurred at 6:21.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 7:26?c) How long (in hours) is the tide (height of the water) 16 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(6π (t− 11

60)37

)+ 14

b) h(7 : 26) = 6.34 feetc) ∆t = 5.29 hoursd) hmin = 5 feet, hmax = 23 feet

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15. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 21 feetoccurred at 0:16 and a low tide of 5 feet occurred at 6:26.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 11:28?c) How long (in hours) is the tide (height of the water) 13 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 8 cos

(6π (t− 4

15)37

)+ 13

b) h(11 : 28) = 19.7 feetc) ∆t = 6.17 hoursd) hmin = 5 feet, hmax = 21 feet

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16. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 25 feetoccurred at 7:17 and a low tide of 3 feet occurred at 13:27.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 16:25?c) How long (in hours) is the tide (height of the water) 17 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 11 cos

(6π (t− 437

60 )37

)+ 14

b) h(16 : 25) = 13.35 feetc) ∆t = 5.08 hoursd) hmin = 3 feet, hmax = 25 feet

17. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 22 feetoccurred at 7:11 and a low tide of 6 feet occurred at 13:25.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 7:28?c) How long (in hours) is the tide (height of the water) 17 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 8 cos

(30π (t− 431

60 )187

)+ 14

b) h(7 : 28) = 21.92 feetc) ∆t = 4.71 hoursd) hmin = 6 feet, hmax = 22 feet

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18. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 24 feetoccurred at 2:28 and a low tide of 4 feet occurred at 8:38.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 4:22?c) How long (in hours) is the tide (height of the water) 14 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 10 cos

(6π (t− 37

15)37

)+ 14

b) h(4 : 22) = 19.67 feetc) ∆t = 6.17 hoursd) hmin = 4 feet, hmax = 24 feet

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19. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 24 feetoccurred at 6:19 and a low tide of 6 feet occurred at 12:33.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 9:25?c) How long (in hours) is the tide (height of the water) 15 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(30π (t− 379

60 )187

)+ 15

b) h(9 : 25) = 15.08 feetc) ∆t = 6.23 hoursd) hmin = 6 feet, hmax = 24 feet

20. The height of the water in a bay varies sinusoidally over time. On a certain day, a high tide of 22 feetoccurred at 6:18 and a low tide of 4 feet occurred at 12:28.a) Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight).b) How high is the water at 9:23?c) How long (in hours) is the tide (height of the water) 16 feet or more?d) Graph the trigonometric function h(t).Note. Round the answers to the nearest hundredth.Answers:

a) h(t) = 9 cos

(6π (t− 63

10)37

)+ 13

b) h(9 : 23) = 13.0 feetc) ∆t = 4.83 hoursd) hmin = 4 feet, hmax = 22 feet

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