section 5.4: modeling with circular...
TRANSCRIPT
![Page 1: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/1.jpg)
Section 5.4: Modeling with CircularFunctions
![Page 2: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/2.jpg)
Circular Motion
Example
A ferris wheel with radius 25 feet is rotating at a rate of 3revolutions per minute, When t = 0, a chair starts at its lowestpoint on the wheel, which is 5 feet above ground. Write amodel for the height h (in feet) of the chair as a function of thetime 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.
![Page 3: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/3.jpg)
Circular Motion
Example
A ferris wheel with radius 25 feet is rotating at a rate of 3revolutions per minute, When t = 0, a chair starts at its lowestpoint on the wheel, which is 5 feet above ground. Write amodel for the height h (in feet) of the chair as a function of thetime 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.
![Page 4: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/4.jpg)
Circular Motion
Example
A ferris wheel with radius 25 feet is rotating at a rate of 3revolutions per minute, When t = 0, a chair starts at its lowestpoint on the wheel, which is 5 feet above ground. Write amodel for the height h (in feet) of the chair as a function of thetime 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.
![Page 5: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/5.jpg)
Circular Motion
Example
A ferris wheel with radius 25 feet is rotating at a rate of 3revolutions per minute, When t = 0, a chair starts at its lowestpoint on the wheel, which is 5 feet above ground. Write amodel for the height h (in feet) of the chair as a function of thetime 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.
![Page 6: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/6.jpg)
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 howlong is a revolution here?
20 seconds.
![Page 7: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/7.jpg)
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 howlong is a revolution here?
20 seconds.
![Page 8: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/8.jpg)
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 howlong is a revolution here?
20 seconds.
![Page 9: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/9.jpg)
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 howlong is a revolution here?
20 seconds.
![Page 10: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/10.jpg)
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 howlong is a revolution here?
20 seconds.
![Page 11: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/11.jpg)
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 howlong is a revolution here?
20 seconds.
![Page 12: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/12.jpg)
Circular Motion
Parameters:
A =
−25B = 2π
20 = π10
C = 0D = 30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 13: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/13.jpg)
Circular Motion
Parameters:
A = −25
B = 2π20 = π
10
C = 0D = 30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 14: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/14.jpg)
Circular Motion
Parameters:
A = −25B =
2π20 = π
10
C = 0D = 30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 15: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/15.jpg)
Circular Motion
Parameters:
A = −25B = 2π
20 = π10
C = 0D = 30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 16: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/16.jpg)
Circular Motion
Parameters:
A = −25B = 2π
20 = π10
C =
0D = 30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 17: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/17.jpg)
Circular Motion
Parameters:
A = −25B = 2π
20 = π10
C = 0
D = 30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 18: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/18.jpg)
Circular Motion
Parameters:
A = −25B = 2π
20 = π10
C = 0D =
30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 19: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/19.jpg)
Circular Motion
Parameters:
A = −25B = 2π
20 = π10
C = 0D = 30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 20: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/20.jpg)
Circular Motion
Parameters:
A = −25B = 2π
20 = π10
C = 0D = 30
Putting this together, we have
h(t) = −25cos( π
10t)+ 30
![Page 21: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/21.jpg)
A Modeling Example
Example
A team of biologists have discovered a new creature in the rainforest. They note the temperature of the animal appears to varysinusoidally over time. A maximum temperature of 125◦
occurs 15 minutes after they start their examination. Aminimum temperature of 99◦ occurs 28 minutes later. The teamwould like to find a way to predict the animal’s temperatureover time in minutes. Your task is to help them by creating agraph of one full period and an equation of temperature as afunction over time in minutes
![Page 22: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/22.jpg)
A Modeling Example
•(15,125)
•(43,99)
So this looks like what kind of curve? A sine curve ...
![Page 23: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/23.jpg)
A Modeling Example
•(15,125)
•(43,99)
So this looks like what kind of curve? A sine curve ...
![Page 24: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/24.jpg)
A Modeling Example
•(15,125)
•(43,99)
So this looks like what kind of curve? A sine curve ...
![Page 25: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/25.jpg)
A Modeling Example
•(15,125)
•(43,99)
So this looks like what kind of curve? A sine curve ...
![Page 26: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/26.jpg)
A Modeling Example
•(15,125)
•(43,99)
So this looks like what kind of curve?
A sine curve ...
![Page 27: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/27.jpg)
A Modeling Example
•(15,125)
•(43,99)
So this looks like what kind of curve? A sine curve ...
![Page 28: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/28.jpg)
A Modeling Example
We need to find the midline, amplitude and period first, whichwe can do from the information here. Then we will worryabout the horizontal shift.
How can we find the amplitude?
125 − 992
=262
= 13
So, if the amplitude is 13, what is the midline?
y = 112
![Page 29: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/29.jpg)
A Modeling Example
We need to find the midline, amplitude and period first, whichwe can do from the information here. Then we will worryabout the horizontal shift.
How can we find the amplitude?
125 − 992
=262
= 13
So, if the amplitude is 13, what is the midline?
y = 112
![Page 30: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/30.jpg)
A Modeling Example
We need to find the midline, amplitude and period first, whichwe can do from the information here. Then we will worryabout the horizontal shift.
How can we find the amplitude?
125 − 992
=262
= 13
So, if the amplitude is 13, what is the midline?
y = 112
![Page 31: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/31.jpg)
A Modeling Example
We need to find the midline, amplitude and period first, whichwe can do from the information here. Then we will worryabout the horizontal shift.
How can we find the amplitude?
125 − 992
=262
= 13
So, if the amplitude is 13, what is the midline?
y = 112
![Page 32: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/32.jpg)
A Modeling Example
We need to find the midline, amplitude and period first, whichwe can do from the information here. Then we will worryabout the horizontal shift.
How can we find the amplitude?
125 − 992
=262
= 13
So, if the amplitude is 13, what is the midline?
y = 112
![Page 33: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/33.jpg)
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 knowA = 13B = π
28
D = 112
![Page 34: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/34.jpg)
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 knowA = 13B = π
28
D = 112
![Page 35: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/35.jpg)
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 knowA = 13B = π
28
D = 112
![Page 36: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/36.jpg)
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 knowA =
13B = π
28
D = 112
![Page 37: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/37.jpg)
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 knowA = 13B =
π28
D = 112
![Page 38: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/38.jpg)
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 knowA = 13B = π
28
D =
112
![Page 39: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/39.jpg)
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 knowA = 13B = π
28
D = 112
![Page 40: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/40.jpg)
A Modeling Example
Now the horizontal shift. Notice the curve doesn’t cross themidline 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 oneto the right.
![Page 41: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/41.jpg)
A Modeling Example
Now the horizontal shift. Notice the curve doesn’t cross themidline 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 oneto the right.
![Page 42: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/42.jpg)
A Modeling Example
So, our final equation therefore is ...
y = 13sin( π
28(x − 1)
)+ 112
![Page 43: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/43.jpg)
A Modeling Example
So, our final equation therefore is ...
y = 13sin( π
28(x − 1)
)+ 112
![Page 44: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/44.jpg)
Another Example
Example
On February 10, 1990, high tide in Boston was at midnight. Thewater level at high tide was 9.9 feet; later, at low tide, it was 0.1feet. Assuming the next high tide is exactly 12 hours later andthat 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 = .1
y = 9.9
12 24
![Page 45: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/45.jpg)
Another Example
Example
On February 10, 1990, high tide in Boston was at midnight. Thewater level at high tide was 9.9 feet; later, at low tide, it was 0.1feet. Assuming the next high tide is exactly 12 hours later andthat 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 = .1
y = 9.9
12 24
![Page 46: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/46.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use?
CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 47: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/47.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? Cosine
Amplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 48: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/48.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude?
4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 49: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/49.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9
Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 50: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/50.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline?
5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 51: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/51.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5
Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 52: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/52.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period?
12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 53: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/53.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 54: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/54.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 55: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/55.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9
B = π6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 56: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/56.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 57: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/57.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0
D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 58: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/58.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 59: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/59.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5
![Page 60: Section 5.4: Modeling with Circular Functionsbtravers.weebly.com/uploads/6/7/2/9/6729909/modeling_with_circular... · Circular Motion Example A ferris wheel with radius 25 feet is](https://reader036.vdocuments.us/reader036/viewer/2022071004/5fc1b75e66abbb50b303f5df/html5/thumbnails/60.jpg)
Another Example
Now let’s determine what we need.
What type of curve should we use? CosineAmplitude? 4.9Midline? 5Period? 12 hours
And the parameters ...
A = 4.9B = π
6
C = 0D = 5
Which gives y = 4.9cos(
π6 t)+ 5