A Ferris wheel has a diameter of 60 feet. When you start at the top of the Ferris wheel, you are 62 feet from the ground. The Ferris wheel completes one rotation in 2 minutes.
1.Create a graph that represents your height relative to the ground as a function of time, using the image below as a guide. Replace the image with your graph.
2. Now create a function that represents your height from the ground as a function of time.
3. Compare the function from the previous practical and the one above. Explain what the two functions have in common and what is different. Specifically mention the amplitude, period, and any transformations to map the first function to the second.
4. If the Ferris wheel stops after 8.5 minutes, how high from the ground would you be? Use the appropriate function from above and show work
5. The Ferris wheel starts again from the position you had in question 4, resetting to time = 0. Create a graph that represents your height relative to the ground as a function of time, using the image below as a guide. Replace the image with your graph.
6.6. Create a new function of the height from the ground as a function of time for the situation in question 5.
I don't know where to start, I learned it like 3 months ago.
To create a graph representing your height relative to the ground as a function of time, we can start by understanding the motion of the Ferris wheel.
1. The Ferris wheel has a diameter of 60 feet, meaning its radius is half of that, which is 30 feet.
2. When you start at the top of the Ferris wheel, you are 62 feet from the ground.
3. The Ferris wheel completes one rotation in 2 minutes.
Now, let's create the graph:
- The x-axis represents time in minutes.
- The y-axis represents height from the ground in feet.
At t = 0 minutes, you are at the top of the Ferris wheel, 62 feet from the ground. As time progresses, your height changes due to the circular motion of the wheel.
To create the graph, we need to determine the height as a function of time. This height can be represented using a sinusoidal function. The general form is:
y = A * sin(B * (x - C)) + D
Where:
- A represents the amplitude (maximum displacement from the midline)
- B represents the frequency (or how many cycles occur within a given time period)
- C represents the phase shift (horizontal displacement of the graph)
- D represents the vertical shift (where the midline of the graph is located)
In this case, let's find the values for A, B, C, and D.
A: The amplitude is the maximum displacement from the midline. In this case, from the ground to the highest point on the Ferris wheel is 62 - 0 = 62 feet. Therefore, A = 62.
B: The frequency is the number of cycles completed within the given time period. Since the Ferris wheel completes one full rotation in 2 minutes, the frequency is 1 cycle per 2 minutes or 1/2 cycles per minute. Therefore, B = 1/2.
C: The phase shift represents the horizontal displacement of the graph. Since we want the graph to start at t = 0 minutes when you are at the top of the Ferris wheel, there is no horizontal displacement. Therefore, C = 0.
D: The vertical shift represents where the midline of the graph is located. In this case, the midline of the graph is at 62 + 0 = 62 feet. Therefore, D = 62.
Now we can create the function representing your height from the ground as a function of time:
h(t) = 62 * sin((1/2) * (t - 0)) + 62
Where h(t) represents your height from the ground at time t.
To compare this function with the previous one and identify their similarities and differences:
- The amplitude (A) remains the same in both functions, as it represents the maximum displacement from the midline.
- The frequency (B) remains the same in both functions as well since it represents the number of cycles completed within a given time period.
- The phase shift (C) is 0 in both functions, indicating that both graphs start at the same point in time (t = 0).
- The vertical shift (D) is also the same in both functions, representing the midline of the graph.
In question 4, when the Ferris wheel stops after 8.5 minutes, we can use the function h(t) = 62 * sin((1/2) * (t - 0)) + 62 to find your height from the ground.
h(8.5) = 62 * sin((1/2) * (8.5 - 0)) + 62
h(8.5) = 62 * sin(4.25) + 62
You can calculate sin(4.25) using a calculator or software:
h(8.5) ≈ 62 * (-0.8202) + 62
h(8.5) ≈ -50.77 + 62
h(8.5) ≈ 11.23 feet
Therefore, after 8.5 minutes, you would be approximately 11.23 feet above the ground.
For question 5, where the Ferris wheel starts again from the position you had at t = 8.5 minutes, you can use the same function h(t) = 62 * sin((1/2) * (t - 0)) + 62 to represent your height.
The graph would start at t = 0 and your height would be 11.23 feet above the ground.
Please note that I am unable to generate or provide images as a text-based AI, so I cannot replace the image with the actual graph. However, you can easily plot the graph using various graphing tools or software, inputting the sine function mentioned above.