Sinusoidal function

Question

Two Ferris wheels are rotating side by side at the Math Fair. The first Ferris wheel has a radius of 7m and makes one complete revolution every 16s. The bottom of the wheel is 1.5m above the ground. Write the equation of a sinusoidal function that models the height of one gondola as it goes around the Ferris wheel assuming that the car starts at the minimum at time zero.

 

(Must Use Radians)

Answer 1

One possible solution is

 

h (t) = 1.5 + 14*sin(pi/16)t

 

The reason is this:

 

Because one complete revolution is every 16s, it means that the period T of this sinusoidal function h(t) is:

 

T = 16 s     Then, after 16 s the gondola is located at the same starting point  (h = 1.5 m)

 

But at a time t =8 s, this gondola is located at the highest height in the wheel=  2*r

were "r" is the radius of one Ferris wheel. It happens only when the argument of this sinusoidal function h(t) is pi/2

(remember that sin(pi/2) = 1)

 

Then, a way to get it is as indicated:   (2*r)*sin(pi/T)t +1.5     or  14*sin(pi/16)t + 1.5

 

To prove it, test the following:   At start time t = 0 s   14*sin(pi/16)*0  +1.5  = 0     h(t) = 1.5 m

                                              At time t = 8 s   14*sin(pi/16)*8 + 1.5 = 14*sin(pi/2) +1.5 = 15.5 m

                                              At time t=16 s   14*sin(pi/16)*16 +1.5 = 14*sin(pi) + 1.5 = 1.5 m

 

I hope it can help you