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 246 feet and initial height is 20 above the ground. The wheel makes a full circle every 15 seconds and has a diameter of 226 feet. Sketch a graph of your height as a rider as a function of time.
a. What is an equation for your graph?

Answer 1

Imagine this. You're at the top of the Ferris wheel, which is 246 feet off the ground. Then, like Spider-Man, you climb straight down to the bottom, climbing in a straight vertical line, down to the bottom of the Ferris wheel. The line you're climbing down is the diameter of the Ferris wheel.

 

The distance between 20 (the initial height) and 246 (highest point) is 226, which is the exact distance of the diameter. This means that we start building an equation as soon as you get on the wheel. Sometimes we're asked to make an equation for the ride a few seconds in, but no, in this case, the initial height also happened to be the lowest point of the ride, so we make an equation for the moment you hop in.

 

The first thing to ask yourself is whether a sine or cosine is better for this. Sines start at zero, while cosines start at their maximum, which is usually 1. However, which a negative sign, cosines can start at -1, which is the minimum. The point is, if the situation is starting at a minimum or maximum, cosines are better. The rider is starting at the minimum wheel height so cosines are better.

 

Acos(Bx - C) + D

 

A is the amplitude. The value of A can be found by thinking of it as half the distance between the maximum and minimum. The maximum, in this case, is 246, and the minimum is 20. Find half the distance between those two numbers.

 

(246 - 20)/2 = 113

 

A is 113, but remember, this graph is starting at a minimum. To make cosine start at a minimum, put a negative sign on the amplitude, so A = -113.

 

To find B, here's a nice formula.

 

B = 2π/Period

 

The period is the horizontal length of one cycle. The horizontal axis of a graph is the x-values, and in this case, x-values represent time. The period would therefore be the amount of time it takes for one complete cycle to occur, which is 15 seconds.

 

B = 2π/15

 

C is called a phase shift. This would be a shift on the x-axis. Recall that x-values represent time, so an x-axis shift would represent how much time you were already on the Ferris wheel in this situation. The graph starts the exact moment you climbed onto the Ferris wheel, so no time has already passed, meaning C = 0.

 

D represents a vertical shift, that simply ensures that the minimum and maximum are where they need to be to make this equation accurate. Look at the equation, with everything plugged in. That includes A = -113, B = 2π/15, and C = 0.

 

Acos(Bx - C) + D

-113cos(2πx/15) + D

 

Without a D term, the maximum of the graph is 113, and the minimum is -113. To get the maximum from 113 to 246 (the maximum of the Ferris wheel), there needs to be an upward shift of 133. Adding 133 to -113 also gives 20, which is the Ferris wheel's minimum. This is how we know D = 133 is the final piece of the puzzle.

 

-113cos(2πx/15) + 133

 

I hope this helps, and good luck with your other trigonometry adventures!