What is the equation of the height of a person in a Ferris wheel in the following case?
As the person rides a Ferris wheel, the height from the ground varies sinusoidally with time. Let t be the number of seconds that have elapsed since the Ferris wheel started. It takes 3 seconds to reach the top, 43 feet above the ground. The wheel makes a revolution once every 8 seconds. The diameter of the wheel is 40 feet.
We have the highest point that a person in the Ferris wheel reaches as 43 feet. The diameter of the Ferris wheel is 40 feet. And the time taken to make a revolution is 8 seconds. It is also given that the highest point is reached in 3 seconds.
The function of the distance from the ground with respect to time is d (t) = 23 - 20*(sin ((3/4)*pi + (2/8)*pi*t))
Here the sine term in the function contributes the value that oscillates from -20 to + 20; the 23 ensures that the distance stays between 3 and 43.
The lowest value that d (t) reaches is when the sine term is -20. Here the height is 23-20 = 3 feet. The highest value is when the sine term is + 20; here it is 20 + 23 = 43 feet. The highest point is reached after 3 seconds and one revolution is completed in 8 seconds.
The value of the function only takes positive values as the height of the person from the ground cannot be negative (else s/he would be underground!)
If you want to know the distance at different values of t, just substitute t with that value and solve the function.