Write a possible equation of your graph of a ferris wheel

Question

Problem:  You are about to begin your ride at the very bottom of the ferris wheel, and rotate counter-clockwise from there. Write a possible equation of the graph with these details:  Diameter is 228ft,  Speed: 14 minutes  per revolution,  initial height off ground: 8 ft, Max height: 236

 

b. How high off the ground are you going to be after 5 minutes?

 

c. How high off the ground are you going to be after 10 minutes?

 

d.How high off the ground are you going to be after 15 minutes?

 

e.If you stay on the ferris wheel for a really long time , list the first 3 times where you will be exactly at 30 ft. off the ground?

c. How high off the ground are you going to be after 10 minutes?

Answer 1

Hi,

 

Because we are going around a circle, the sine function is a natural fit.  We can choose the origin of our coordinate system at any point.  Choosing the center of the ferris wheel simplifies the math.  Since we are given the period of rotation (T = 14 min.), we can calculate the angular velocity as 2π/T = 2π/14 = π/7.

 

We begin to build our formula with: y=sin((π/7)t)  

 

For every 14 minutes, the argument of the sine function sweeps through 2π of arc (or one full rotation) from 0 at t = 0 to 2π at t = 14.  This meets the criteria that the wheel make one rotation in 14 minutes.  But notice that the sine function has a range of (-1,1).  We are going to ride up 114 ft (half the given diameter) above the origin at the top and 114 ft below the origin at the bottom.  We need to multiply the result by 114.

 

Now we have:  y = 114 sin(πt/7)

 

This formula has a range of (-114, 114).  We want our distance from the ground, not the origin.  Since the origin is 114+8 above the ground, we must add 122 to equation.

 

This yields y = 114 sin (πt/7) + 122

 

Finally, we start the ride at the bottom of the loop.  At t = 0, the equation we have written yields y = 122.  We want it to yield y = 8 at the start.  The sine function yields -1 at -π/2.  Subtracting π/2 from πt/7 inside the argument of the sine function would correct this.  

 

Our final solution is; y = 114 sin ((πt/7)-π/2) + 122

 

Check this by solving at 1/4 turn increments, t = 0, 3.5, 7, 10.5, and 14.

 

At t = 5 min., 

 

y= 114 sin ((5π/7)-π/2)+122 = 114 sin ((10π/14) - (7π/14)) + 122 = 114 sin (3π/14) +122

 

y ≅ 193 ft