As you ride a ferris wheel, your distance from the ground varies sinusoidally with time. When the last passenger boards the ferris wheel and the ride starts moving, let your position be modeled by the diagram provided.

Question

As you ride a ferris wheel, your distance from the ground varies sinusoidally with time. When the last passenger boards the ferris wheel and the ride starts moving, let your position be modeled by the diagram provided.

Let t be the number of seconds that have elapsed since you began moving. It takes you 5 seconds to reach the top of the wheel (38 feet above the ground) and 20 seconds to make a complete revolution. The diameter of the wheel is 30 feet.
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Write the equation of a sinusoid that describes the motion?
A=15
B: pi/10
C:??? (phase shift)
D:??? (Vertical shift)

Answer 1

C:none

D: 8 feet

Answer 2

D is actually 23 bc its 8 ft above the lowest point, so you also need to add the 15 feet of radius

Answer 3

To find the equation of a sinusoid that describes the motion, we need to use the following general form:

y = A*sin(B(x - C)) + D

In this equation:
- A represents the amplitude (half the distance between the maximum and minimum values) of the sinusoid.
- B represents the coefficient inside the parentheses, which affects the horizontal stretch or compression of the graph.
- C represents the phase shift of the graph.
- D represents the vertical shift of the graph.

Given the information provided:
- The amplitude is given as A = 15.
- The time taken to reach the top of the wheel (maximum height) is 5 seconds, and the height is 38 feet. So the maximum value of the wave is 38 feet, and the distance from the ground to the center of the wheel is 38 - 15 = 23 feet.
- The diameter of the wheel is 30 feet, so the radius is 15 feet, which is equal to the amplitude.
- The time taken to complete one revolution is 20 seconds. Since the circumference of the wheel is given by C = 2πr = 2π(15), the distance traveled in 20 seconds is 2π(15). Therefore, the coefficient B can be calculated as B = 2π(15) / 20 = π/2.

Now let's find the values of phase shift (C) and vertical shift (D):
- The phase shift represents the horizontal translation of the graph. Since the ferris wheel starts at its lowest point, the phase shift is 0 seconds or 0 units to the right (initial position).
- The vertical shift represents the vertical translation of the graph. Since the ferris wheel is initially at a height of 23 feet from the ground, the vertical shift is 23 feet.

Now we have all the required values:

A = 15
B = π/2
C = 0
D = 23

Therefore, the equation of the sinusoid that describes the motion is:

y = 15*sin(π/2(x - 0)) + 23

Simplifying further:

y = 15*sin(πx/2) + 23

The correct answer would be B: π/10 for B (coefficient inside the parentheses) and D: 23 for D (vertical shift).

Answer 4

To write the equation of a sinusoid that describes the motion, we can use the general form of a sinusoidal function:

y = A * sin(B(x - C)) + D

Where:
- A represents the amplitude of the function, which is half the difference between the maximum and minimum values. In this case, the amplitude is given as 15.
- B represents the frequency of the function, which is determined by how quickly the function repeats. The period of a sinusoid can be calculated as T = 2π/B, so if it takes 20 seconds to make a complete revolution, the period is 20 seconds. Using the formula, we can determine B = 2π/T.
- C represents the phase shift of the function, which determines the horizontal translation. It tells us where the function begins. We are not given the phase shift in the information provided.
- D represents the vertical shift of the function, which moves the entire graph up or down. It tells us the vertical position when the ferris wheel starts moving.

By analyzing the information given, we can determine:
- The amplitude is A = 15.
- The period T is 20 seconds, so B = 2π/20 = π/10.
- The vertical shift D is the position at t = 0 seconds, which is not specified, so we cannot determine it.
- The phase shift C, which determines the starting point of the motion, is also not provided in the given information.

Thus, we are missing the values for both the vertical shift (D) and the phase shift (C) to write the full equation of the sinusoid. The answer choices provided (B: π/10 and C: ???) suggest that there might be a typo in the question or that important information is missing.