Let f(t) denote your height (in meters) above ground at t minutes. Find a formula for f(t).

Question

A ferris wheel is 35 meters in diameter and boarded at ground level. The wheel makes one full rotation every 4 minutes, and at time you are at the 9 o'clock position and descending. Let  denote your height (in meters) above ground at  minutes. Find a formula for f(t). 

Answer 1

Hi Caroline,

If we consider the ground level to be h = 0, then your height above the ground can be thought of as the y-coordinate of the circle. We use sin(t) to denote the y-coordinate of a circle, so we'll start with that.

f(t) = A*sin(Bt - C) + D

|A| is the amplitude. The amplitude of a sine wave is half the distance between the crest and the trough. Since the Ferris Wheel has a diameter of 35 meters, the amplitude will be 35/2 = 17.5. Since this Ferris Wheel is moving clockwise (at 9 o'clock position its descending) A = -17.5

B helps you determine your period. The period of a sine wave is 2π/B. The question says that a full rotation occurs every 4 minutes, so the period is 4. That means 4 = 2π/B, so B = π/2.

C helps you determine your phase shift. The phase shift of a sine wave = C/B. If we consider "t" to be the time after the wheel is at the 9 o'clock position, then the phase shift would be 0. C, therefore, is 0.

D is your vertical shift. Since we start 17.5 feet above ground level at 9 o'clock, D = 17.5.

Your function, then, is:

f(t) = -17.5*sin(π/2*t) + 17.5

Hope this helps!

Answer 2

To model periodic behavior, we will use a sinusoidal function, sine or cosine, of the form f(t) = ± Asin(B(x - C)) + D. For this given information, sine is easier and we can have C = 0, as explained below.

Because the 9 o-clock starting position is the average height of the ferris wheel (even, vertically, with the center), we use the sine function. And because the seat is descending at time = 0, it will be a - sine.

A represents the amplitude of the function, and is = to the radius of the wheel, 17.5.

D represents the vertical shift and because the center is 17.5 m above ground level, D = 17.5.

B controls the period of the function, and the relationship is given by B = 2π / period. So, B = 2π/4 = π/2.

Lastly, because the seat starts at the average height, no phase (horizontal) shift is required, thus C = 0.

Putting this altogether we have the following ...

t: time (in mins) ; f(t): vertical height above ground (in m)

f(t) = - 17.5 sin(π/2t) + 17.5