A Ferris wheel has a diameter of 60 feet. When you start at the bottom of the Ferris wheel, you are 2 feet from the ground.The Ferris wheel completes one rotation in 2 minutes.

Create a graph that represents your height relative to the center of the Ferris wheel as a function of time.
Create a function that represents your height relative to the center of the Ferris wheel as a function of time

well, let's see...

cos(x) starts at the top, and has amplitude 1.

You want to start at the bottom, with amplitude 30.

cos(kx) has period 2π/k. You want period 2. So, what is k?

To create a graph that represents your height relative to the center of the Ferris wheel as a function of time, we can use a sinusoidal function, since the Ferris wheel completes one rotation in 2 minutes.

The general equation for a sinusoidal function is:
y = A sin(Bt + C) + D

Where:
A is the amplitude (half the distance between the highest and lowest points)
B is the period (the length of time it takes to complete one cycle)
C is a phase shift (the horizontal shift of the graph)
D is the vertical shift (the average value)

In this case, since the diameter of the Ferris wheel is 60 feet, the radius (amplitude) is 60/2 = 30 feet. The period is 2 minutes, and since the function is based on time, the phase shift C is 0. The vertical shift D is the distance from the ground, which is 2 feet.

Therefore, the function for your height relative to the center of the Ferris wheel as a function of time is:
y = 30 sin((2π/2)t) + 2

This equation represents your height above the ground (relative to the center of the Ferris wheel) at any given time t.

To create a graph that represents your height relative to the center of the Ferris wheel as a function of time, we need to understand the motion of the Ferris wheel.

Given that the diameter of the Ferris wheel is 60 feet, we can calculate the radius using the formula: Radius = Diameter / 2. Therefore, the radius of the Ferris wheel is 60 / 2 = 30 feet.

Now, let's break down the motion of the Ferris wheel into its components. The Ferris wheel completes one rotation in 2 minutes, which means it completes 360 degrees of rotation in 2 minutes.

To model your height relative to the center of the Ferris wheel as a function of time, we can use a cosine function since it represents periodic motion. The cosine function oscillates between -1 and 1, producing a wave-like pattern.

In this case, the amplitude of the cosine function will be the radius of the Ferris wheel, which is 30 feet. We can use the following formula to represent your height (h) as a function of time (t):

h(t) = A * cos(ωt) + d

Where:
- A represents the amplitude of the cosine function, which is the radius of the Ferris wheel (30 feet).
- ω (omega) represents the angular frequency, which is calculated by dividing 2π (the number of radians in one complete rotation) by the time it takes for one full rotation (2 minutes or 120 seconds).
- t represents the time in seconds.
- d represents the initial vertical displacement, which is the distance between your starting point and the center of the Ferris wheel (2 feet).

Substituting the values into the equation, we get:

h(t) = 30 * cos((2π/120) * t) + 2

Now, we can create a graph using this equation to show your height relative to the center of the Ferris wheel as a function of time.