A Ferris wheel is 40 meters in diameter and boarded from a platform that is 5 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes one full revolution every 7 minutes. You make two complete revolutions on the wheel, starting at t = 0.
Graph h = f(t), where h is the height above the ground (in meters) at time t, in minutes. Indicate the period, the amplitude, and the midline of your graph. [Hint: first determine an appropriate interval for t, with t ≥ 0.]
r = 20
period = 7 min = T
when t = 0
h = 5 at t = 0 and wheel does rotation between h = 5 and h = 45
center at h = 25
so of form
h = 25 - 20 cos (2 pi t/T )
To graph h = f(t) for the given scenario, we can follow these steps:
Step 1: Determine the appropriate interval for t
Since the Ferris wheel completes one full revolution every 7 minutes, we can choose an interval of t = [0, 14]. This ensures that we cover two complete revolutions on the wheel, as required.
Step 2: Identify the period
The period of a function is the distance required for one full cycle or revolution. In this case, the period of the Ferris wheel is 7 minutes because it completes one full revolution in that time.
Step 3: Determine the midline
The midline represents the average value of the function. In this case, the midline is the height of the loading platform, which is 5 meters above the ground level.
Step 4: Find the amplitude
The amplitude represents the maximum distance from the midline. For a Ferris wheel, the amplitude is half the diameter of the wheel. In this case, the diameter is 40 meters, so the amplitude is 20 meters.
Step 5: Plot the graph
On a coordinate plane, let the x-axis represent time (t) in minutes, and the y-axis represent height (h) in meters.
For the given interval, we can plot points on the graph using the height formula for a Ferris wheel: h = A * sin((2π/P) * t) + midline.
In this case, the amplitude (A) is 20 meters, the period (P) is 7 minutes, and the midline is 5 meters.
Using these values, we can calculate the corresponding heights at different times and plot them on the graph.
The graph of the function h = f(t) will have a sinusoidal shape, with the points repeating every 7 minutes.
Note: To create a precise graph, it is useful to use a graphing calculator or software.
I hope this helps! Let me know if you have any further questions.
To graph the function h = f(t), we need to understand the behavior of the Ferris wheel and determine the height above the ground at different points in time.
Let's break down the problem step by step:
1. Determine an appropriate interval for t: Given that the Ferris wheel completes one full revolution every 7 minutes and you make two complete revolutions, the appropriate interval would be from t = 0 to t = 14 minutes.
2. Determine the period: The period is the time it takes for the graph to complete one full cycle. In this case, since you make two complete revolutions in 14 minutes, the period is 14 minutes.
3. Determine the amplitude: The amplitude is the maximum displacement from the midline of the graph. Since the Ferris wheel has a diameter of 40 meters, the radius (and hence the amplitude) is half of that, which is 20 meters.
4. Determine the midline: The midline is the average value of the graph. In this case, the midline would be 5 meters above the ground, as stated in the problem.
To graph the function h = f(t), we will use a sine function because it represents periodic motion.
The general form of the equation for the sine function is h = A*sin(B(t - C)) + D, where A is the amplitude, B determines the period (B = 2π/period), C represents horizontal shifts (phase shifts), and D is the vertical shift.
In this case, the equation becomes h = 20*sin(2π/14(t - 0)) + 5, simplifying to h = 20*sin((π/7)t) + 5.
Now, plot the graph with t on the x-axis and h on the y-axis. Use the interval we determined earlier (t = 0 to t = 14 minutes) to plot points on the graph.
By plugging in different values of t between 0 and 14 into the equation, you can find the corresponding height above the ground (h).
For example:
- When t = 0 minutes, h = 20*sin((π/7)*0) + 5 = 20*sin(0) + 5 = 0 + 5 = 5 meters
- When t = 7 minutes, h = 20*sin((π/7)*7) + 5 = 20*sin(π) + 5 = 20*0 + 5 = 5 meters (the same as the midline)
- When t = 14 minutes, h = 20*sin((π/7)*14) + 5 = 20*sin(2π) + 5 = 20*0 + 5 = 5 meters (the same as the midline)
Using these values, plot the points on the graph and connect them to form a smooth curve. The resulting graph will have a period of 14 minutes, an amplitude of 20 meters, and a midline of 5 meters.
Note: The graph would be a sinusoidal curve that repeats every 14 minutes, starting and ending at the midline (y = 5).