Section 5.4: Modeling with Circular Functions

Transcription

1 Section 5.4: Modeling with Circular Functions

2 Circular Motion Example A ferris wheel with radius 25 feet is rotating at a rate of 3 revolutions per minute, When t = 0, a chair starts at its lowest point on the wheel, which is 5 feet above ground. Write a model for the height h (in feet) of the chair as a function of the time t (in seconds).

3 Circular Motion Example A ferris wheel with radius 25 feet is rotating at a rate of 3 revolutions per minute, When t = 0, a chair starts at its lowest point on the wheel, which is 5 feet above ground. Write a model for the height h (in feet) of the chair as a function of the time t (in seconds). What kind of curve should we use - sine or cosine?

4 Circular Motion Example A ferris wheel with radius 25 feet is rotating at a rate of 3 revolutions per minute, When t = 0, a chair starts at its lowest point on the wheel, which is 5 feet above ground. Write a model for the height h (in feet) of the chair as a function of the time t (in seconds). What kind of curve should we use - sine or cosine? First, let s find the amplitude. Thoughts?

5 Circular Motion Example A ferris wheel with radius 25 feet is rotating at a rate of 3 revolutions per minute, When t = 0, a chair starts at its lowest point on the wheel, which is 5 feet above ground. Write a model for the height h (in feet) of the chair as a function of the time t (in seconds). What kind of curve should we use - sine or cosine? First, let s find the amplitude. Thoughts? Since the radius is 25 feet, the amplitude is 25 feet.

6 Circular Motion What is the maximum of the chair?

7 Circular Motion What is the maximum of the chair? Minimum?

8 Circular Motion What is the maximum of the chair? Minimum? What is the midline?

9 Circular Motion What is the maximum of the chair? Minimum? What is the midline? y = 30

10 Circular Motion What is the maximum of the chair? Minimum? What is the midline? y = 30 Now the period... we are looking for t to be in seconds, so how long is a revolution here?

11 Circular Motion What is the maximum of the chair? Minimum? What is the midline? y = 30 Now the period... we are looking for t to be in seconds, so how long is a revolution here? 20 seconds.

12 Circular Motion Parameters: A =

13 Circular Motion Parameters: A = 25

14 Circular Motion Parameters: A = 25 B =

15 Circular Motion Parameters: A = 25 B = 2π 20 = π 10

16 Circular Motion Parameters: A = 25 B = 2π 20 = π 10 C =

17 Circular Motion Parameters: A = 25 B = 2π 20 = π 10 C = 0

18 Circular Motion Parameters: A = 25 B = 2π 20 = π 10 C = 0 D =

19 Circular Motion Parameters: A = 25 B = 2π 20 = π 10 C = 0 D = 30

20 Circular Motion Parameters: A = 25 B = 2π 20 = π 10 C = 0 D = 30 Putting this together, we have h(t) = 25cos ( π 10 t ) + 30

21 A Modeling Example Example A team of biologists have discovered a new creature in the rain forest. They note the temperature of the animal appears to vary sinusoidally over time. A maximum temperature of 125 occurs 15 minutes after they start their examination. A minimum temperature of 99 occurs 28 minutes later. The team would like to find a way to predict the animal s temperature over time in minutes. Your task is to help them by creating a graph of one full period and an equation of temperature as a function over time in minutes

22 A Modeling Example So this looks like what kind of curve? A sine curve...

23 A Modeling Example So this looks like what kind of curve? A sine curve...

24 A Modeling Example (15,125) So this looks like what kind of curve? A sine curve...

25 A Modeling Example (15,125) (43,99)

26 A Modeling Example (15,125) (43,99) So this looks like what kind of curve?

27 A Modeling Example (15,125) (43,99) So this looks like what kind of curve? A sine curve...

28 A Modeling Example We need to find the midline, amplitude and period first, which we can do from the information here. Then we will worry about the horizontal shift.

29 A Modeling Example We need to find the midline, amplitude and period first, which we can do from the information here. Then we will worry about the horizontal shift. How can we find the amplitude?

30 A Modeling Example We need to find the midline, amplitude and period first, which we can do from the information here. Then we will worry about the horizontal shift. How can we find the amplitude? = 26 2 = 13

31 A Modeling Example We need to find the midline, amplitude and period first, which we can do from the information here. Then we will worry about the horizontal shift. How can we find the amplitude? = 26 2 = 13 So, if the amplitude is 13, what is the midline?

32 A Modeling Example We need to find the midline, amplitude and period first, which we can do from the information here. Then we will worry about the horizontal shift. How can we find the amplitude? = 26 2 = 13 So, if the amplitude is 13, what is the midline? y = 112

33 A Modeling Example Now, the period... thoughts?

34 A Modeling Example Now, the period... thoughts? It takes 28 minutes from the highest to the lowest temperature, so the period therefore is

35 A Modeling Example Now, the period... thoughts? It takes 28 minutes from the highest to the lowest temperature, so the period therefore is 56 minutes.

36 A Modeling Example Now, the period... thoughts? It takes 28 minutes from the highest to the lowest temperature, so the period therefore is 56 minutes. So at this point we know A =

37 A Modeling Example Now, the period... thoughts? It takes 28 minutes from the highest to the lowest temperature, so the period therefore is 56 minutes. So at this point we know A = 13 B =

38 A Modeling Example Now, the period... thoughts? It takes 28 minutes from the highest to the lowest temperature, so the period therefore is 56 minutes. So at this point we know A = 13 B = π 28 D =

39 A Modeling Example Now, the period... thoughts? It takes 28 minutes from the highest to the lowest temperature, so the period therefore is 56 minutes. So at this point we know A = 13 B = π 28 D = 112

40 A Modeling Example Now the horizontal shift. Notice the curve doesn t cross the midline on the y-axis, so there is a horizontal shift. (15,125) (43,99) How can we find how big this shift needs to be?

41 A Modeling Example Now the horizontal shift. Notice the curve doesn t cross the midline on the y-axis, so there is a horizontal shift. (15,125) (43,99) How can we find how big this shift needs to be? Based on length of period and where peaks are, the shift is one to the right.

42 A Modeling Example So, our final equation therefore is...

43 A Modeling Example So, our final equation therefore is... ( π ) y = 13sin 28 (x 1) + 112

44 Another Example Example On February 10, 1990, high tide in Boston was at midnight. The water level at high tide was 9.9 feet; later, at low tide, it was 0.1 feet. Assuming the next high tide is exactly 12 hours later and that the height of the water is given by a sine or cosine curve, find a formula for water level in Boston as a function of time t.

45 Another Example Example On February 10, 1990, high tide in Boston was at midnight. The water level at high tide was 9.9 feet; later, at low tide, it was 0.1 feet. Assuming the next high tide is exactly 12 hours later and that the height of the water is given by a sine or cosine curve, find a formula for water level in Boston as a function of time t. y = y =.1

46 Another Example Now let s determine what we need. What type of curve should we use?

47 Another Example Now let s determine what we need. What type of curve should we use? Cosine

48 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude?

49 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9

50 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline?

51 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5

52 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period?

53 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period? 12 hours

54 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period? 12 hours And the parameters...

55 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period? 12 hours And the parameters... A = 4.9

56 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period? 12 hours And the parameters... A = 4.9 B = π 6

57 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period? 12 hours And the parameters... A = 4.9 B = π 6 C = 0

58 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period? 12 hours And the parameters... A = 4.9 B = π 6 C = 0 D = 5

59 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period? 12 hours And the parameters... A = 4.9 B = π 6 C = 0 D = 5

60 Another Example Now let s determine what we need. What type of curve should we use? Cosine Amplitude? 4.9 Midline? 5 Period? 12 hours And the parameters... A = 4.9 B = π 6 C = 0 D = 5 Which gives y = 4.9cos ( π 6 t) + 5