Tony visits the local fair and sees one if therides, 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'?

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To find the equation of the height of a person on the Ferris wheel at any time 't', we need to consider the motion of the Ferris wheel as it rotates.

The height of a person on the Ferris wheel is determined by the vertical component of their position. We can use the equation of a circle to represent the height as a function of time.

First, let's define some variables:
- r: radius of the Ferris wheel (half the diameter)
- h: height of the person on the Ferris wheel
- t: time in seconds
- T: time taken to complete one full revolution (given as 12 seconds)

From the diameter provided, we can calculate the radius:
r = 50 feet / 2 = 25 feet

Now, we can derive the equation for the height of a person on the Ferris wheel at time 't'.

1. Determine the angle of rotation θ at time 't':
Since the Ferris wheel takes 12 seconds to complete one full revolution, the angle θ covered per second is:
θ = (2π radians) / T = (2π radians) / 12 seconds = π / 6 radians/second
The angle at time 't' can be found as:
θ = (π / 6) * t

2. Calculate the vertical component of the person's position h:
Using basic trigonometry, we can find the vertical component:
h = r * sin(θ)
Note that the sine function provides the vertical component given an angle.

Now, we have the equation for the height of a person on the Ferris wheel at any time 't' (in feet):
h = 25 * sin((π / 6) * t)

This equation will give you the height of a person on the Ferris wheel at any given time 't'.