A ferris wheel has diameter of 24 m and is 4 m above ground level at its lowest point. Assume that a rider enters a car from a platform that is 30^0 around the rim before the car reaches its lowest point. Model the rider's height above the ground versus angle using a transformed sine function

If we consider x the angle traveled since getting on the wheel,

h = 16+12sin(x-120)

Thanks Steve, but what is the reason for minus 120 and not plus 120?

I drew the diagram with the person boarding at -120°. As the wheel turns counter-clockwise, x starts increasing from 0°. Depending on how you set up your diagram, maybe +120° is appropriate.

To model the rider's height above the ground versus angle using a transformed sine function, we need to consider the equations of sinusoidal functions and the given information about the ferris wheel.

The equation of a basic sine function is:

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

Where:
- A determines the amplitude (half the difference between the maximum and minimum values),
- B determines the period (the horizontal distance between two consecutive peaks or troughs),
- C determines the phase shift (horizontal shift), and
- D determines the vertical shift (the midline or average value).

In this case, we need to adjust the basic sine function to meet the given conditions. Let's go step-by-step:

Step 1: Determine the amplitude (A)
The amplitude is described as half the difference between the maximum and minimum values. In this case, the ferris wheel is 4 m above ground level at its lowest point, so the amplitude is 4/2 = 2 m.

Step 2: Determine the period (B)
The period represents the horizontal distance between two consecutive peaks or troughs. Since the rider enters the car from a platform that is 30° around the rim before reaching the lowest point, we need to determine the angle at which the rider reaches the lowest point.

In a full circle, there are 360°, and the rider enters the car at a position 30° before the lowest point. So, the rider reaches the lowest point at 360° - 30° = 330°.

Since the period is the distance in which the function completes one full cycle, we can calculate it using the formula: period = 360° / B. In this case, we know that the rider reaches the lowest point at 330°, so

360° / B = 330°

We can solve this equation to find the value of B:

B = 360° / 330°

Step 3: Determine the phase shift (C)
The phase shift is the horizontal shift of the waveform. In this case, the rider is already 30° into the cycle when they enter the car, so the phase shift is 30°.

Step 4: Determine the vertical shift (D)
The vertical shift represents the midline or average value of the waveform. Since the ferris wheel starts 4 m above ground level at its lowest point, the vertical shift D is 4 m.

Putting it all together, the transformed sine function can be represented as:

y = 2 sin((360° / 330°)(x - 30°)) + 4

This equation models the rider's height above the ground versus angle using a transformed sine function.