Write an equation for a Ferris wheel using the given numbers

Question

You begin your ride at the very bottom of the Ferris Wheel, and rotate counter-clockwise from there. Write an equation for the Ferris Wheel given the numbers that were given to you.

 

Diameter (feet): 256

 

Speed (minutes per revolution): 19

Initial Height (feet off of the ground): 14

 

Max Height (feet): 270

 

 

 

Answer 1

This is a favorite example of a certain Trigonometry textbook author, but I won't mention his name.

 

I am assuming that what you want is the height of a rider on the wheel in terms of the time. If that is not correct, you may need to make some adjustments.

 

Here is a recap of the given information.

 

The bottom of the wheel is 14 ft above ground.

The radius of the wheel is 256/2 = 128 ft.

The angular velocity (or speed) is 1/19 rpm.

 

To begin, the angular velocity should be converted to a more useful unit. Since 1 revolution = 2π radians, you can write it as

 

ω = 2π/19 . (The letter on the left side of the = is a Greek letter called "omega." It is the last letter in the Greek alphabet.) The units on this number are radians per minute.

 

Next, after t minutes, the wheel will rotate through the angle θ = ωt radians. That is θ = (2π/19) t radians.

 

How do you model the height of the rider. Here is a hint. When t = 0, the rider is at his/her lowest point, with a height of 14 ft. After a half-revolution (θ = π rads), he/she is at the top of the wheel, a height of 270 ft.

 

The best function to use will be a negative cosine.

 

Try using h = k + A cos (Bt) as the model for the height. When t = 0, h = 14. This means that

 

14 = k + A.

 

The radius of the wheel will give the amplitude, so A = -128 and the height of the axle of the wheel is 128+14 = 142 ft, so k = 142. This agrees with this equation, since 14 = 142 - 128.

 

The next step is to find B. Remember that the period of the cosine graph is 2π/B. Since the wheel rotates once in 19 minutes, the period is 19 min. Thus,

 

2π/B = 19 ⇒ B = (2π)/19 .

 

The equation (all put together) is

 

h = 142 - 128 cos ((2πt)/19) .

 

As a check on this, try putting t = 0 in this and verify that h = 14 ft. Try putting t = 19/2 min the verify that h = 270 ft. Also, try putting in t = 19 min and verify that h = 14 ft again.

 

It works!

 

Question: If you wanted the height in terms of the angle and not the time, how would you change this equation?