Write a Sinusoidal function for the following. A ferris wheel is 100 feet in diameter and makes one revolution in 45 seconds. If the rider needs to walk up a ramp to get to their seat that is 5 feet above the ground, write the model for the rider.
To write a sinusoidal function for the rider on a ferris wheel, we need to consider the height of the rider at any given time.
Let's start by finding the amplitude of the function. The diameter of the ferris wheel is 100 feet, which means the radius is half of that, or 50 feet. Since the rider needs to walk up a ramp that is 5 feet above the ground, the maximum height of the rider on the ferris wheel would be 50 + 5 = 55 feet. Therefore, the amplitude of our function would be 55 feet.
Next, we need to determine the period of the function, which is the time it takes for the ferris wheel to complete one full revolution. Given that the ferris wheel makes one revolution in 45 seconds, the period of our function is 45 seconds.
Now, let's express the sine function in the form:
h(t) = A * sin(2π / T * t + ϕ) + k
Where:
- A is the amplitude
- T is the period
- t is the time variable
- ϕ is the phase shift
- k is the vertical shift (in this case, the height of the ramp, which is 5 feet)
Substituting the values we have:
h(t) = 55 * sin(2π / 45 * t + ϕ) + 5
The phase shift, ϕ, determines where the function starts. Since the rider starts from the ground, the phase shift should be 0. Therefore, our final equation becomes:
h(t) = 55 * sin(2π / 45 * t) + 5
This equation represents the height of the rider on the ferris wheel as a function of time.