Sinusoidal Function (Word Problem)

Question

At an amusement park, the Ferris wheel has a diameter of 42 ft, and it is positioned 5 feet above the ground. To ride costs 7 tickets, so Charlie wants to time the ride to make sure he gets his money’s worth. Once all the passengers are loaded, the ride begins, and Charlie starts his stopwatch. He reaches the top 9 seconds later and returns to the top when the stopwatch reads 33 seconds, and again at 57 seconds. While on the Ferris wheel, Charlie’s height above the ground (h, in feet) is a sinusoidal function of time (t, in seconds).

A. What is the period of this function? Why

B. Write a function of the form h(t) = a cos(b(t -c)) + d to model this situation.

Answer 1

Hello Dorothy,

A. By definition, the amount of time between two repeated events is the period. The problem tells you Charlie reaches the top 9 seconds after starting his stopwatch, then at 33 seconds and then again at 57 seconds.

How many seconds have gone by between 9 and 33? How many seconds have gone by between 33 and 57 seconds?

That answer will be the period of this function.

B.

a = amplitude = (peak value reached - lowest value)/2

If the ride begins at the bottom of the Ferris wheel, you're still 5 feet off the ground, so lowest value = 5

At the peak, Charlie will be 5 feet off the ground PLUS the diameter of the wheel, so he'll be 47 feet off the ground. So amplitude = a = (47-5)/2 = 21

b = (2π/period). Since you will have found the period from question A, you just plug it in here.

d = midline = (peak value + lowest value)/2 = (47+5)/2 = 52/2 = 26

For c, you're asked to give an equation using cosine. By definition, the cosine function starts a cycle at the top, then to the midline, then the bottom, midline, then back to the top.

The function described in the problem does not reach the top until 9 seconds AFTER Charlie started his stopwatch. That means the natural "behavior" of cosine is delayed by 9 seconds. so 9 will be the value of c.

You now will have found a, b, c and d. You can then plug them in the equation and you've solved the problem.

Cheers!