Example: A Ferris wheel has a radius of 15 meters and the bottom rests 1 meter above the ground level. If the Ferris wheel makes one complete revolution every 45 seconds, find an equation for the height of a person from the ground as they ride the Ferris wheel. Use \(y\) for the height of the person and \(t\) be the time in seconds. Finally, assume that the rider in question is 16 meters above the ground at time zero and going down.
Solution First, let's consider what "1 revolution per 45 seconds" means. If you pick any point on the Ferris wheel, after 45 seconds that point will complete one full revolution. Hence, the point returns to the same position and continues in the same direction as when we started observing 45 seconds ago. This is quite literally the definition of the period of a function. Next, let's consider what the average height of someone riding the Ferris wheel is. Given the radius is 15, it would make sense to say that 15 is the average height, however, the problem also states that the Ferris wheel is 1 meter above ground, so the height ends up being 16 meters. The final description indicates that we are observing someone starting at a height of 16 meters (the average) and that they are traveling down initially. Starting at the average value indicates this is best described with the Sine function, and going down means that it has a negative sign. Lastly, the amplitude is the radius since that dictates precisely how high and low the person travels. Putting all of this together:
\[ \solve{ y &=& -15\sin(Bt)+16\\ 45&=&\dfrac{2\pi}{{B}}\\ B&=&\dfrac{2\pi}{{45}}\\ y&=&\boxed{-15\sin\left(\frac{2\pi}{{45}}t\right)+16 } }\]