You are in a car of a ferris wheel. The wheel has a radius of 8m and turns counterclockwise. Let the orgin be at the center of the wheel. Begin each sketch in parts a) and b) when the radius from the centre of the wheel to your car is along the positive x-axis.
a) Sketch the graph of your horizontal displacement versus the angle through which you turn for one rotation of the wheel. Which function models the horizontal displacement? Justify your choice.
b) Sketch the graph of your vertical displacement versus the angle through which you turn for one rotation of th wheel. Which function models the vertical displacement?Justify your answer.
Last one:) Sorry...:(
We cannot sketch on these posts.
a) To sketch the graph of horizontal displacement versus the angle through which you turn, we can consider that the horizontal displacement is the distance along the x-axis from the origin.
As given, the radius from the center of the wheel to your car is along the positive x-axis. This means that at the starting point, your horizontal displacement is 8 meters (equal to the radius).
Since the wheel turns counterclockwise, as you rotate, your horizontal displacement decreases until it reaches a minimum of -8 meters when you are at the opposite side of the circle. Then, as you continue rotating, your horizontal displacement increases again until you reach the starting point.
Therefore, the graph of your horizontal displacement versus the angle will look like an oscillating sine wave centered at 0. The function that models this horizontal displacement is a sine function of the form:
f(θ) = A sin(θ)
where A is the amplitude (which is 8 in this case), and θ represents the angle through which you turn.
b) To sketch the graph of vertical displacement versus the angle through which you turn, we can consider that the vertical displacement is the distance along the y-axis from the origin.
Since the car is on a ferris wheel, the vertical displacement only changes as the wheel turns. At the starting position, your vertical displacement is 0, as you are aligned with the center of the wheel. As you rotate counterclockwise, your vertical displacement increases until it reaches a maximum of 8 meters when you are at the topmost point of the wheel. Then, as you continue rotating, your vertical displacement decreases until it reaches 0 again when you are aligned with the center of the wheel.
Therefore, the graph of your vertical displacement versus the angle will look like a simple up and down motion. The function that models this vertical displacement is a cosine function of the form:
g(θ) = B cos(θ)
where B is the amplitude (which is 8 in this case), and θ represents the angle through which you turn.
Note: The amplitude in both cases is equal to the radius of the wheel, which is 8 meters, because it represents the maximum displacement from the center of the wheel.
No problem! I can help you with that. To sketch the graphs and determine the functions that model the horizontal and vertical displacements, we need to understand the relationship between the angle through which you turn and the corresponding displacements.
a) To sketch the graph of your horizontal displacement versus the angle through which you turn, we need to consider that as the ferris wheel rotates counterclockwise, your car moves along a circular path. The radius of the ferris wheel is 8m, which means that as you rotate through an angle θ, your horizontal displacement from the origin can be found using the formula:
Horizontal Displacement = radius * cos(θ)
In this case, since the radius from the center of the wheel to your car is along the positive x-axis at the start, the angle θ represents the counterclockwise rotation and is measured from the positive x-axis.
Now, let's sketch the graph. Since the ferris wheel completes one full rotation (360° or 2π radians), we can vary the angle θ from 0 to 2π and calculate the corresponding horizontal displacements using the above formula. Plotting these pairs of values on a graph will give us the desired graph of horizontal displacement.
b) Similarly, to sketch the graph of your vertical displacement versus the angle through which you turn, we need to consider that your car's vertical displacement from the origin depends on the angle θ as well. The formula for vertical displacement is:
Vertical Displacement = radius * sin(θ)
Again, since the radius from the center of the wheel to your car is initially along the positive x-axis, the angle θ represents counterclockwise rotation and is measured from the positive x-axis.
Using the same approach as before, we can vary the angle θ from 0 to 2π and calculate the corresponding vertical displacements using the above formula. Plotting these pairs of values on a graph will give us the desired graph of vertical displacement.
By examining these graphs, we can determine the functions that model the horizontal and vertical displacements. The function for horizontal displacement is given by:
f(θ) = 8 * cos(θ)
And the function for vertical displacement is given by:
g(θ) = 8 * sin(θ)
These functions model the relationship between the angle through which you turn and the corresponding horizontal and vertical displacements.