Write an equation about the movement of a Ferris wheel.

Question

As a Ferris wheel turns , the distance a rider is above the ground varies sinusoidally with time.  The highest point on the wheel is 43 feed above the ground.  The wheel makes a full circle every 8 seconds and has a diameter of 40 feet. Sketch a graph of your height as a rider as a function of time.

a.  What is an equation for your graph?

b.  Use this equation to predict your height above the ground at 4.5 seconds.

 

 

Answer 1

The general form of a sine function is y = A sin (Bt+C) + D

A is the amplitude, the amount that the sine travels above and below its midpoint

2π/B is the period of the graph: the amount of time it takes to complete one cycle.

-C/B is the phase shift, how far away from 0 the sine graph begins.

D is the vertical shift, how far above or below zero the midpoint of the graph is.

 

In this problem, the rider starts at the minimum point of the cycle. This is most like a cosine graph that is flipped over. You can write this by making the amplitude negative.

 

The maximum height is 43, and the diameter is 40, so the minimum height is 3. The total distance from top to bottom is 40, so the amplitude A is 40/2 = 20 ft. Since we're inverting a cosine, A = -20 ft.

The problem states that the period is 8 seconds, so B = 2π/8 = π/4.

Cosines start at the minimum, so there is no phase shift. C = 0.

The center of the Ferris Wheel is minimum + A = 3+20 = 23 ft = D.

 

Your equation is therefore:

y = -20 cos(π t/4) + 23

 

Substitute t=4.5 to solve for part b.

 

(Note: I did this problem in radians. If you want to use degrees, then the equation for B is period = 360/B. A and D don't depend on how you measure the angle.)