A ferris wheel has a diameter of 10 m and takes 24 sec to make one revolution. The lowest point on the wheel is 1 m above the ground. (4 marks) a. Sketch a graph to show how a rider's height above the ground varies with time as the ferris wheel makes a rotation. Assume the person starts the ride at the lowest point on the wheel.
b. Write a trigonometric equation that describes the graph.
c. Check the accuracy of your equation by using t = 12 sec. Explain why this provides a check of the accuracy of the equation. What are two more values that would provide a good check?
surely you can solve at least some of these exercises. At least the amplitude?
Review the material about sinusoidal functions, and I think some of it will become clear.
Recall that
A sin(Bx)
has amplitude A and period 2pi/B.
Also, look at the graph of cos(x). It has a max at t=0. You want a min at t=0.
a. To sketch a graph showing how a rider's height above the ground varies with time, we can assume that the center of the ferris wheel is at the origin (0,0). The rider starts at the lowest point on the wheel, which is 1 m above the ground. As the wheel rotates, the rider travels in a circular path. We can represent the equation of the circle using the equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
Since the center of the circle is at (0,0), the equation becomes:
x^2 + y^2 = r^2
The radius (r) of the circle is equal to half the diameter, which is 10 m / 2 = 5 m.
The rider's height (y) above the ground will vary sinusoidally as the ferris wheel rotates. To determine the equation that describes this variation, we need to consider the starting position and the period of the ferris wheel.
At the starting position, when the rider is at the lowest point 1 m above the ground, the equation becomes:
x^2 + (y - 1)^2 = 25
The period of the ferris wheel is the time it takes for one complete revolution, which is given as 24 seconds. Since the equation of a circle has a period of 2π, we need to scale it accordingly. The period of the sinusoidal variation will be 24 seconds / (2π radians) ≈ 3.82 seconds/radian.
b. To write a trigonometric equation that describes the graph, we can express the variation in height (y) as a function of the angle (θ) the rider has traveled along the ferris wheel. This can be done using the sine function since it represents the vertical component of a point moving in a circular path.
The equation can be written as:
y = 1 + 5sin(3.82θ)
c. To check the accuracy of the equation using t = 12 seconds, we need to find the corresponding angle (θ) that the rider has traveled in that time. We can use the equation:
θ = (2π/period) * t
θ = (2π/24) * 12 ≈ 2.094 radians
Substituting this value in the equation gives us:
y = 1 + 5sin(3.82 * 2.094) ≈ 1 + 5sin(7.979) ≈ 4.91 m
This means that after 12 seconds, the rider would be approximately 4.91 m above the ground.
Two more values that would provide a good check are t = 0 seconds and t = 24 seconds.
For t = 0 seconds, the equation gives us:
θ = (2π/24) * 0 = 0 radians
y = 1 + 5sin(0) = 1 m
This confirms that when the rider starts the ride, they are at the lowest point, 1 m above the ground.
For t = 24 seconds, the equation gives us:
θ = (2π/24) * 24 = 2π radians
y = 1 + 5sin(3.82 * 2π) ≈ 1 + 5sin(24) ≈ 1 m
This confirms that after one complete revolution, the rider returns to the starting position, at 1 m above the ground.
a. To sketch the graph, we need to determine the height of the rider above the ground at different points in time. Let's break down the information given:
- The diameter of the ferris wheel is 10 m. This means the radius is half of the diameter, so the radius is 10 / 2 = 5 m.
- The lowest point on the wheel is 1 m above the ground.
- The ferris wheel takes 24 sec to make one revolution.
With this information, we can visualize the graph. At the start of the ride (t = 0 sec), the rider is at the lowest point, 1 m above the ground. As time progresses, the rider moves up and then down. At t = 12 sec, the rider will be at the highest point, and at t = 24 sec, the rider will be back at the lowest point.
To sketch the graph, we can plot the time (t) on the x-axis and the height above the ground (h) on the y-axis. Connect the points to show the variation in height as the ferris wheel completes a rotation.
b. To write the trigonometric equation that describes the graph, we can use the cosine function. The cosine function describes the variation in height over time as the rider moves up and down on the ferris wheel.
Here's how we can write the trigonometric equation:
h = A * cos(B * t) + C
Where:
- A is the amplitude, which is half the range of variation in height. From the given information, the amplitude is (5 - 1) / 2 = 2 m.
- B is the frequency, which is determined by the time it takes for one complete cycle. In this case, the ferris wheel takes 24 sec to complete one revolution, so the frequency is 2π / 24 = π / 12.
- C is the vertical shift, which is the average height of the rider above the ground. In this case, the lowest point is 1 m above the ground, so C = 1.
Therefore, the trigonometric equation that describes the graph is:
h(t) = 2 * cos((π / 12) * t) + 1
c. To check the accuracy of the equation, we can substitute t = 12 sec into the equation and compare the result with the expected height at that time. The expected height at t = 12 sec is the highest point of the ferris wheel, which is 5 m above the ground. If the calculated height matches the expected height, it confirms the accuracy of the equation.
For t = 12 sec:
h(12) = 2 * cos((π / 12) * 12) + 1
= 2 * cos(π)
= 2 * (-1) (since cos(π) = -1)
= -2
Since the calculated height is -2 m, which is not equal to the expected height of 5 m, we can conclude that there might be an error in our equation.
To further check the accuracy of the equation, we can use additional values of t. Two more values that would provide a good check are t = 0 sec and t = 24 sec.
For t = 0 sec:
h(0) = 2 * cos((π / 12) * 0) + 1
= 2 * cos(0)
= 2 * 1 (since cos(0) = 1)
= 2
For t = 24 sec:
h(24) = 2 * cos((π / 12) * 24) + 1
= 2 * cos(2π)
= 2 * 1 (since cos(2π) = 1)
= 2
Since the calculated heights for t = 0 sec and t = 24 sec match the expected heights of 2 m and 1 m respectively, it provides a good check of the accuracy of the equation.