Question

Ralph gets on a Ferris wheel at the 6 o'clock position which is located 5 feet off of the ground. Once in motion, the Ralph's distance feom the ground varies from 5 feet to 25 feet off of the ground every 30 seconds. ( After 30 seconds he is at the 12 o'clock position)
a) Sketch a graph of Ralph's distance D from the ground over a 1 minute interv as a function of time t. Assume t = 0 corresponds to a time wheb Ralph was closed to the ground. Remember to label your axis.
b) Determine what the period, midline, amplitude, and horizontal phase shift.​

Answer 1

The distance D from the ground follows a sine law, which begins at D=5 where t=0 and reaches a maximum of D=25 where t=30.

This implies that the sine function has an amplitude of 25-5=20, and a period of 30 seconds.

The resulting function is

$20\sin\left(\dfrac{\pi t}{60}\right)+5$

In fact, this function:

Has a value of 5 where t=0: $20\sin(0)+5=0+5=5$

Reaches the maximum at t=30: $20\sin\left(\frac{\pi}{2}\right)+5 = 20+5=50$

Resets after 60 seconds: $20\sin\left(2\pi\right)+5 = 20\sin(0)+5 = 0+5=5$

So, the period is 60 seconds, the midline is 15 (the middle value between 5 and 25), the amplitude is 20, and there is no horizontal phase shift.