2002 AMC 10B Problems/Problem 24
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point vertical feet above the bottom?
We can let this circle represent the ferris wheel with center and represent the desired point feet above the bottom. Draw a diagram like the one above. We find out is a triangle. That means and the ferris wheel has made of a revolution. Therefore, the time it takes to travel that much of a distance is of a minute, or seconds. The answer is . Alternatively, we could also say that is congruent to by SAS, so is 20, and is equilateral, and
Solution 2 (trig)
The path that the rider takes along the Ferris wheel can be represented by a sinusoidal graph, where represents the time in seconds. Since is at the crest of the graph and not at the midline, we will use a cosine graph. Therefore, we will use the form:
The graph starts at the lowest point at feet, then goes up to reach the highest point at feet, then comes back down. Therefore, the amplitude is (and negative since it starts at the bottom, not the top), and the vertical shift is . There is no horizontal shift since the lowest point is at . It takes seconds to make one full revolution (the period), so Now we have all the parts we need for the equation of our graph, and we can set it equal to the height we want,
We get to and remember that the problem is looking for the first instance of , so Solving, we get that .
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