Tony visits the local fair and sees one of the rides, the ferris wheel. The ferris wheel has a diameter of 50 feet and is on a platform of 4 feet. If it takes 12 seconds to make one full revolution, what is the equation of the height of a person on the ferris wheel at any time t?

assuming the person starts at the wheel's lowest position when t=0, that means that f(t) will look something like

f(t) = -cos(t)

The radius is 25 feet, so that makes it

f(t) = -25cos(t)

The axle is 25+4=29 feet off the ground, so

f(t) = 29-25cos(t)

since cos(kt) has period 2π/k, we have 2π/k = 12, so

f(t) = 29-25cos(π/6 t)

We could use either a sine or a cosine function, you did not specify which, but I will use sine

It must be of the type
height = a sin k(Ø + d) + c
period = 12 s
= 2π/k
2π/k = 12
k = 2π/12 = π/6

also we know a = 25

so far we have
height = 25 sin π/6(t + d) + c

you did not say where you want the person to be when t = 0 , that will determine the phase shift

you did not say how the height you want relates to the 4 foot platform. Do you want your height to describe the height above ground or above the platform.

To find the equation of the height of a person on the ferris wheel at any time t, we can start by figuring out the height of the person when the ferris wheel is at its lowest point (or ground level).

The lowest point of the ferris wheel is at a height of 4 feet above the ground, which is the height of the platform.

Next, we need to find the maximum height that the person reaches when the ferris wheel is at its highest point.

The ferris wheel has a diameter of 50 feet, which means its radius is half of the diameter, so it is 50/2 = 25 feet.

When the person is at the highest point, they are at a height of 25 feet above the ground (the radius of the ferris wheel).

Now, let's consider one revolution of the ferris wheel. It takes 12 seconds to complete one full revolution.

Since the height of the person on the ferris wheel varies sinusoidally over time, we can express the height as a function of time using a sine function.

The general form of a sine function is:
h(t) = A*sin(B(t - C)) + D

In this case, A represents the amplitude (half the vertical distance between the highest and lowest points), B represents the frequency (the number of cycles per unit of time), C represents the phase shift (a horizontal shift of the waveform), and D represents the vertical shift (the average height level).

For the given ferris wheel problem:
- The amplitude (A) is half the vertical distance between the highest and lowest points, which is (25 - 4)/2 = 21.5 feet.
- The frequency (B) is the reciprocal of the time it takes to complete one full revolution, which is 1/12 cycles per second.
- The phase shift (C) is 0 since we're considering time t to be measured from the start.
- The vertical shift (D) is the lowest point of the ferris wheel, which is 4 feet.

Putting it all together, the equation of the height of a person on the ferris wheel at any time t is:
h(t) = 21.5*sin((1/12)(t - 0)) + 4
Simplified, it becomes:
h(t) = 21.5*sin((t/12)) + 4

This equation gives the height of a person on the ferris wheel at any time t.