The figure on the left shows a Ferris wheel with a dot representing a boy, Tommy, riding it. The Ferris wheel started revolving when Tommy was at the bottom. The figure on the right shows the graph 𝑦 equals 𝑑 of 𝑡. Which of the following statements could describe the function of time when the Ferris wheel started revolving? A) The speed at which the Ferris wheel is revolving, B) the distance between Tommy and the ground, C) the number of revolutions Tommy made through his ride, or D) the distance between Tommy and the center of the Ferris wheel.
We see the dot that represents Tommy on this Ferris wheel. And now, we need to consider which of the four situations best describes this graph. A says the speed at which the Ferris wheel is revolving. If this is true, then the 𝑦-axis represents speed. As the time increases, the longer Tommy is on the ride. Could this line represent the speed of the Ferris wheel? Let’s describe what would be happening along this line. We would have an increase in speed and then a decrease in speed; an increase in speed, a decrease in speed; and an increase in speed again.
At first, we might think that this is a viable option for a Ferris wheel. However, a few details should make us reconsider. We should notice these minimum points on our graph. They never cross the 𝑥-axis. And crossing the 𝑥-axis would represent a speed of zero. In this function, 𝑦 never equals zero. The speed never equals zero. And this means that the Ferris wheel never stops. And in most Ferris wheels, the machine would come to a complete stop to let passengers on and off. The only way this graph could be true, as if passengers had to enter and exit the Ferris wheel while it was moving. So let’s consider another option.
Option B is the distance between Tommy and the ground. If the 𝑦-axis represents the distance from the ground, Tommy moves further and further away from the ground, then goes closer to the ground, away from the ground again, closer to the ground, and away from the ground. Again, we should consider these dips, these minimum points. Because we know that 𝑦 does not equal zero on this graph. If the graph represents the distance between Tommy and the ground, it’s telling us that while Tommy is on the Ferris wheel, he never touches the ground. And this would make sense because the Ferris wheel doesn’t drag the ground. The lowest point on the Ferris wheel still maintains some distance from the ground. And this means option B is a possibility.
What about option C, the number of revolutions Tommy made through his ride? The 𝑦-axis would be the number of revolutions. And for this to be true, we immediately see a problem. This graph has two portions that are decreasing. And Tommy can never go backwards in the number of revolutions he’s completed. The longer he stays on the Ferris wheel, the more revolutions he completes. A graph of the revolutions Tommy completed would have to start at zero, zero because before he started, he would have completed zero revolutions and would have to increase at a consistent rate. The longer he stays on, the more revolutions he completes. Option C can’t describe the function we’ve been given.
Finally, we have option D, the distance between Tommy and the center of the Ferris wheel. This Ferris wheel is a circle. And the distance between Tommy and the center of the circle could be considered a radius of this circle. As Tommy goes around and around this Ferris wheel, he maintains the exact distance from the center, no matter where he is along the route. Graphing his distance from the center of the Ferris wheel as a function of time would look like this. His distance from the center doesn’t change the longer he’s on the wheel.
And so, we say that the best description for the graph 𝑦 equals 𝑑 𝑡 is the distance between Tommy and the ground, option B.