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 center of the Ferris wheel as a function of time.
2. Create a function that represents your height relative to the center of the Ferris wheel as a function of time. (15 points)
the amplitude is 60/2 = 30
The axle is 32 feet up
The period of cos(kx) is 2pi/k
1. To create a graph that represents your height relative to the center of the Ferris wheel as a function of time, we can plot the height on the y-axis and the time on the x-axis. The Ferris wheel completes one rotation in 2 minutes, so we can divide the time axis into 2-minute intervals.
Starting from the top of the Ferris wheel (62 feet from the ground), your height will decrease as you move downward. At the bottom of the Ferris wheel, your height will be -2 feet (62 - 60 = -2). As you continue to move upward, your height will increase until you reach the top of the wheel again.
2. To create a function that represents your height relative to the center of the Ferris wheel as a function of time, we can use a trigonometric function to model the vertical motion. Specifically, we can use the cosine function.
Let's assume that at time t = 0, you are at the top of the Ferris wheel. With one rotation completed in 2 minutes, the angular speed (ω) of the Ferris wheel can be calculated as 2π radians per 2 minutes, which simplifies to π radians per minute.
Based on the given information, the equation that represents your height (h) relative to the center of the Ferris wheel as a function of time (t) can be written as:
h(t) = r * cos(ωt) + d
Where:
- r is the radius of the Ferris wheel (half of the diameter), which is 30 feet.
- ω is the angular speed of the Ferris wheel, which is π radians per minute.
- t is the time in minutes.
- d is the initial height from the ground, which is 62 feet.
Therefore, the function that represents your height relative to the center of the Ferris wheel as a function of time is:
h(t) = 30 * cos(πt) + 62.
To create a graph that represents the height relative to the center of the Ferris wheel as a function of time, we need to understand the motion of the Ferris wheel.
1. The Ferris wheel has a diameter of 60 feet, which means its radius is half of that, or 30 feet.
2. When you start at the top of the Ferris wheel, you are 62 feet from the ground. This means the height from the center of the Ferris wheel to the top is 62 feet.
3. The Ferris wheel completes one rotation in 2 minutes.
To create the graph, we can focus on the height above the center of the Ferris wheel as a function of time. Let's assume that at time t = 0, you start at the top of the Ferris wheel.
Since the Ferris wheel completes one rotation in 2 minutes, it means that it completes 1/2 rotation (180 degrees) in 1 minute. This means that every minute, the Ferris wheel moves 180 degrees.
To find the height at any given time, we can use the concept of circular motion and trigonometry. The height above the center of the Ferris wheel can be represented by the formula:
h(t) = r * sin(θ)
Where:
- h(t) is the height above the center of the Ferris wheel at time t.
- r is the radius of the Ferris wheel.
- θ is the angular position of the Ferris wheel at time t.
In this case, we know that the radius of the Ferris wheel is 30 feet. To find the angular position at any given time, we can use the equation:
θ = (360 * t) / (2 * minutes)
Where:
- θ is the angular position of the Ferris wheel in degrees.
- t is the time in minutes.
Therefore, the function representing your height relative to the center of the Ferris wheel as a function of time can be written as:
h(t) = 30 * sin((360 * t) / (2 * minutes))
To create a graph, plot the time (t) on the x-axis and the height (h) on the y-axis.