Writing Sinusoidal Equations

You ride a Ferris Wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. Suppose the diameter of a Ferris Wheel is 42 feet and travels at a rate of 3 revolutions per minute. At the highest point, a seat on the Ferris wheel is 46 feet above the ground. a. Write a sinusoidal equation to model the height of your seat if you got on the Ferris Wheel at its lowest point. b. According to the model, after you have ridden

the Ferris Wheel for 4 minutes and 13 seconds, the Ferris Wheel comes to an abrupt stop. What would be the height of your seat at that time?

= revolution per 10 seconds

0 se

c

2.5

sec

7.5

sec

12.5

sec

17.5

sec

20 s

ec

5 se

c

10 s

ec

15 s

ec

3 revolutions per min. == 1 revolution per 20 seconds

12

= revolution per 5 seconds= revolution per 2.5 seconds1

8

14

Let’s look at the height every2.5 seconds

0 se

c

2.5

sec

7.5

sec

12.5

sec

17.5

sec

20 s

ec

5 se

c

10 s

ec

15 s

ec

-- 4 ft. --

-- 46 ft. --

Period: 20 seconds

Vertical Shift: 25

Amplitude: 21

2BPeriod

2 20B

10 B

cosy A Bx C D

Phase Shift: 0

1021cos 25y x

A = -21

D = 25

C = 0

21-- 25 ft. --

0 se

c

2.5

sec

7.5

sec

12.5

sec

17.5

sec

20 s

ec

5 se

c

10 s

ec

15 s

ec

-- 4 ft. --

-- 46 ft. --

1021cos 25y x

21-- 25 ft. --

What is the height at 4 minutes and 13 seconds?

1021cos 253 25y

37.343 feet

4(60) + 13 = 253 seconds

(5, 32)

(9, -8)

Write the equation of a sine function.

(7, 12) 20

Period: 8

Vertical Shift: 12

Amplitude: 20

2BPeriod

2 8B

4 B

Phase Shift: 7

A = -20

D = 12

420sin 12y x C

4 7 0C 74C

74 420sin 12y x

(15, 12)

siny A Bx C D

4 0x C

The center of a piston on a windmill tower is 80 meters above the ground. The blades on the windmill are 30 meters in length and turn at a rate of 2 revolutions per minute. The windmill has 3 blades and at the tip of each blade is a light. Two of the lights are white and the 3rd light is blue. We can use a sinusoidal equation to model the height of the light compared to time. If we begin tracking the path of the blue light when the light is at its lowest position at 12:00 midnight (use this as your 0 time), write a sinusoidal equation relating the height to time. Then use your equation to determine the height of the light at 12:10 a.m.

= revolution per 15 seconds

0 se

c

3.75

sec

11.2

5 se

c

18.7

5 se

c

26.2

5 se

c

30 s

ec

7.5

sec

15 s

ec

22.5

sec

2 revolutions per min. == 1 revolution per 30 seconds

12

= revolution per 7.5 seconds= revolution per 3.75 seconds1

8

14

Let’s look at the height every3.75 seconds

0 se

c

3.75

sec

11.2

5 se

c

18.7

5 se

c

26.2

5 se

c

30 s

ec

7.5

sec

15 s

ec

22.5

sec

-- 50 m-

-- 110 m--

Period: 30 seconds

Vertical Shift: 80

Amplitude: 30

2BPeriod

2 30B

15 B

cosy A Bx C D Phase Shift: 0

1530cos 80y x

A = -30

D = 80

C = 0

30-- 80 m --

0 se

c

3.75

sec

11.2

5 se

c

18.7

5 se

c

26.2

5 se

c

30 s

ec

7.5

sec

15 s

ec

22.5

sec

-- 50 m-

-- 110 m--

1530cos 80y x

30-- 80 m --

Now, determine the height of the light at 2:10 a.m.2 10min 7800sec.hr

1530cos 7800 80y

109.155 m