A Ferris wheel has a diameter of 100 feet. When you start at the bottom of the Ferris wheel, you are 2 feet from the ground. The Ferris wheel completes one rotation in 4 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.
since it starts at a minimum when t=0, it will have the form
y = a - b cos(kt)
so apply what you know about sines and cosines, to get
y = 52 - 50 cos(π/2 x)
1. I apologize, but as a text-based AI, I am unable to directly create and display images or graphs. However, I can describe to you how the graph should look like based on the given information.
For the graph, the x-axis represents time in minutes, and the y-axis represents your height from the ground in feet.
The graph should be a sine wave that starts at 2 feet above the ground and goes up to 102 feet when you reach the top of the Ferris wheel, and then comes back down to 2 feet as you reach the bottom.
When the Ferris wheel completes one rotation in 4 minutes, the graph should have one complete cycle in 4 minutes. This means that from the starting position at the bottom, it takes 2 minutes to reach the top, and another 2 minutes to come back down to the bottom.
Therefore, the graph should start at 2 feet on the y-axis, rise to 102 feet at the 2-minute mark, descend back to 2 feet at the 4-minute mark, and repeat this cycle.
2. To create a function that represents your height from the ground as a function of time, we can use a sine function. Let's denote the time in minutes as t.
The function can be written as:
h(t) = (max_height - min_height) * sin(2π / T * t) + min_height
Where:
- max_height is the maximum height from the ground, which is 102 feet in this case.
- min_height is the minimum height from the ground, which is 2 feet in this case.
- T is the time it takes for one complete rotation of the Ferris wheel, which is 4 minutes in this case.
- 2π is the angle in radians for one complete circle.
Substituting the values into the function, we get:
h(t) = (102 - 2) * sin(2π / 4 * t) + 2
Simplifying:
h(t) = 50 * sin(π/2 * t) + 2
Therefore, the function that represents your height from the ground as a function of time is h(t) = 50 * sin(π/2 * t) + 2.
To create a graph representing your height relative to the ground as a function of time, let's break down the problem:
1. First, let's consider the period of the Ferris wheel. The Ferris wheel completes one rotation in 4 minutes, which means it takes 4 minutes for one complete cycle.
2. Next, let's determine the highest point on the Ferris wheel. The diameter of the Ferris wheel is 100 feet, so the radius (half of the diameter) is 50 feet. Considering that you're starting at a height of 2 feet from the ground, the highest point would be at a height of 50 + 2 = 52 feet.
3. Now, we need to identify the lowest point of the Ferris wheel. Since you start at a height of 2 feet from the ground, the lowest point would be 50 - 2 = 48 feet.
4. Our next step is to find the amplitude of the function, which is the distance between the highest and lowest points. In this case, the amplitude would be (52 - 48) / 2 = 2 feet.
5. Finally, we need to determine whether the function starts from the highest or lowest point at time t = 0. Since the problem states that you start at the bottom of the Ferris wheel, which is 2 feet from the ground, the function will start from the lowest point.
Based on these considerations, the function representing your height from the ground as a function of time, h(t), can be represented as a sinusoidal function:
h(t) = A * sin(2π / T * t) + C
Where:
- A is the amplitude
- T is the period
- C is the vertical shift (in this case, the average of the highest and lowest points)
In our case:
A = 2 (amplitude)
T = 4 (period)
C = (52 + 48) / 2 = 50 (vertical shift)
Therefore, the function h(t) can be written as:
h(t) = 2 * sin(2π / 4 * t) + 50
Now, utilizing this function, you can create your graph representing your height relative to the ground as a function of time.