As the Ferris Wheel Turns From Trigonometric Ratios to Trigonometric Functions, Part 2

Adapted from High Dive, a unit from Year 4 of the Interactive Mathematics Program published by Key Curriculum Press (2000)

The Ferris Wheel You have always been afraid of heights, and now your friends have talked you into taking a ride on the amusement park Ferris wheel. As you wait nervously in line you have been able to gather some information about the wheel. By asking the ride operator, you have found out that this wheel has a radius of 25 feet, and its center is 30 feet above the ground. You have also been carefully timing the rotation of the wheel and have found that it makes one complete revolution counterclockwise every 24 seconds. Because the wheel has twelve spokes, you realize that you can describe various positions in the cycle of the Ferris wheel in terms of the face of a clock, as indicated in the accompanying diagram. For example, the highest point in the wheel’s cycle is the 12 o’clock position, and the point farthest to the right is the 3 o’clock position. With this information, you are trying to figure out how high you will be at different positions on the wheel. For simplicity, you think of your location as simply a point on the circumference of the wheel’s circular path.   1. Find your height above the ground at different positions on the wheel. 2. You realize that if you time the ride you will be able to decide at what times you will feel comfortable glancing at the surrounding scenery, and at what times you will need to stare intently at your watch. Find your height above the ground at different times t, where t represents the elapsed time after you pass the 3 o’clock position.  You may want to create various representations–such as a table, a graph or an equation– illustrating the relationship between the elapsed time and your height above ground. How would these representations change if:

x the radius of the wheel was larger or smaller? x the height of the center of the wheel was greater or smaller? x the wheel rotates faster or slower?

How does your strategy or representation help you find your height above ground at specific instances in time, such as t = 5 seconds, t = 13 seconds, or t = 7.5 seconds? To describe your horizontal position, you could use a horizontal coordinate system in which an object’s x-coordinate is based on its distance (in feet) to the right or left of the center of the Ferris wheel, with objects to the right of the center having positive x-coordinates. For instance, in this coordinate system you would have an x-coordinate of 25 when you are at the 3 o’clock position, and an x-coordinate of -25 when you are at the 9 o’clock position. 3. Find your horizontal position at different times t, where t represents the elapsed time after you pass the 3 o’clock position.  Again, you may want to create various representations of this relationship.