Answer: To write an equation that represents the path of the Ferris wheel, we need to consider its circular motion as it rotates. The Ferris wheel is essentially moving in a circular path, and its height varies as a function of time.
Let's consider the height of a point on the Ferris wheel above the ground at any given time t. We'll call this height "h" (in meters).
The center of the Ferris wheel is placed 2 meters above the ground. So, the highest point of the Ferris wheel (when it's at the topmost position) will be at 2 meters + 42/2 meters = 23 meters above the ground. This is the starting height when the person gets on the ride.
As the Ferris wheel rotates, the height of the person varies as a sinusoidal function due to its circular motion. We can model this as a sine function.
Let's assume the Ferris wheel completes one full rotation every T seconds. We are given that T = 36 seconds.
Since the maximum height is 23 meters and the minimum height (when it's at the bottommost position) will be 2 meters above the ground, the amplitude of the sine function will be (23 - 2)/2 = 21/2 = 10.5 meters.
The time period (T) of one full rotation is 36 seconds, which corresponds to one full cycle of the sine function.
The equation that represents the path of the Ferris wheel is given by:
h(t) = A * sin(2π * t / T) + C,
where:
h(t) is the height above the ground at time t,
A is the amplitude of the sine function (half the difference between the maximum and minimum heights),
sin is the sine function,
2π is the radian measure for one full cycle of the sine function,
t is the time in seconds,
T is the period of one full rotation (36 seconds in this case),
C is the vertical shift or the starting height of the Ferris wheel (23 meters in this case).
Substitute the values into the equation:
h(t) = 10.5 * sin(2π * t / 36) + 23.
This equation represents the height of a point on the Ferris wheel above the ground at any given time t as it rotates.