Many carnivals have a version of the double Ferris wheel. A large central arm rotates clockwise. At each end of the central arm is a Ferris wheel that rotates clockwise around the arm. Assume that the central arm has length 200 feet and rotates about its center. Also assume that the wheels have radius 40 feet and rotate at the same speed as the central arm.Find parametric equations for the position of a rider and graph the rider's path. Adjust the speed of rotation of the wheels to improve the ride
The central arm's endpoints are a circle of radius 100:
x = 100cosθ
y = 100sinθ
At θ=0, the arm is horizontal. So, assuming the wheel's boarding position is also at θ=0, and its angle is relative to the central arm, the boarding location's coordinates are
x = 100cosθ + 40cos2θ
y = 100sinθ + 40sin2θ
Not sure what you mean by "improve the ride."
To find the parametric equations for the position of a rider on the double Ferris wheel, we can use the polar coordinates system. Let's assume that the angle the central arm makes with the positive x-axis is θ, and the angle each of the Ferris wheels makes with the positive x-axis is φ.
The position of the rider can be described using the distance r from the origin (center of the central arm) and the angles θ and φ. We can express r and φ in terms of θ to obtain the parametric equations.
1. For the central arm:
The angle of the central arm θ changes as it rotates, so we can relate θ to the length s along the central arm using the equation:
s = rθ
We know that the length s is equal to the length of the central arm, which is 200 feet. Therefore:
200 = rθ
From this equation, we can solve for r in terms of θ:
r = 200/θ
2. For the Ferris wheels:
The radius of the Ferris wheels is given as 40 feet. Therefore, the distance from the center of the central arm to the Ferris wheels is 40 feet. This distance can be expressed in terms of r and φ as:
r = 40 + 40*cos(φ)
To find the position of the rider, we combine the equations for r and φ:
r = 200/θ
r = 40 + 40*cos(φ)
Now, we have the parametric equations for the position of a rider on the double Ferris wheel:
x = r*cos(θ) = (200/θ)*cos(θ)
y = r*sin(θ) = (200/θ)*sin(θ)
To graph the rider's path, we can choose values for θ within the desired range and calculate the corresponding x and y coordinates using the parametric equations. By connecting these points in a continuous line, we will obtain the rider's path on the double Ferris wheel.
As for adjusting the speed of rotation to improve the ride, it depends on the specific criteria for improvement. Increasing or decreasing the speed of rotation of the wheels can affect the overall experience of the riders in terms of thrill, comfort, or other factors. Adjusting the speed is a subjective matter and would require further analysis or specific requirements.
To find parametric equations for the position of a rider on the Ferris wheel, let's break it down step by step:
1. Define the coordinate system: Let's assume we have a Cartesian coordinate system with the origin at the center of the Ferris wheel. The x-axis will be the line connecting the centers of both wheels, and the positive direction will be towards the end where the central arm is rotating.
2. Parameterize the position of the rider on the central arm: We can use an angle θ (in radians) to represent the position of the rider on the central arm. As the arm rotates, the angle θ increases, starting from θ = 0 at the positive x-axis.
3. Parameterize the position of the rider on each wheel: Since the wheels are rotating around the central arm, we can also use angles to represent the position of the rider on each wheel. Let's call the angle for the left wheel α and the angle for the right wheel β. As the central arm rotates, these angles also change.
4. Define the position of the rider: We can find the position of the rider by considering the distances from the origin to the rider on each wheel. We know the lengths of the central arm and the radius of the wheels. Let's call the distance from the origin to the rider for the left wheel d₁ and for the right wheel d₂.
5. Express d₁ and d₂ in terms of θ, α, β: Since the central arm has a length of 200 feet, d₁ can be described as d₁ = 200 - 40cos(θ), where the cosine term accounts for the vertical displacement caused by the rotation of the arm. Similarly, d₂ can be described as d₂ = 200 + 40cos(θ).
6. Substitute α and β in terms of θ: We need to relate α and β to θ since they are also changing as the central arm rotates. From the diagram, we can observe that α = θ - π/2 and β = θ + π/2.
7. Obtain the parametric equations: By substituting the expressions for α, β, d₁, and d₂ into the Cartesian coordinate system, we can obtain the parametric equations for the rider's position:
x = d₁cos(α) = (200 - 40cos(θ))cos(θ - π/2)
y = d₁sin(α) = (200 - 40cos(θ))sin(θ - π/2)
Graphing the rider's path will provide a visual representation of the equations. By adjusting the speed of rotation of the wheels, you can change the shape and characteristics of the path. Experimenting with different speeds can help find a balanced and enjoyable ride experience.