The figure above represents two tangential wheels touching one another at exactly one point. The larger wheel has a circumference of 6π and the smaller wheel has a circumference of 4π. If x and y are the centers of the wheels, what is the distance from point x to point y?
A. 5 B. 6 C. 10 D. 12 E. 15
(A) This question is essentially just asking for the sum of the two radii, since the wheels are touching.
Since circumference = 2πr, radius = circumference/2π. Thus, the smaller wheel has a radius of 4π/2π = 2 and the larger wheel has a radius of 6π/2π = 3.
The distance between x and y is 2 + 3 = 5. The correct answer is choice (A). ---------- What is the difference between the number of revolutions the two gears make per minute?
Though this information is irrelevant to this particular question, let us analyze what happens when wheels rotate.
Imagine the both wheels roll on a line (connected as they are). When the larger one makes a full turn, the wheels will travel the distance of its circumference, or 6π. Therefore the smallest wheel will turn once to travel 4π, plus half a turn to travel 2π. (4π + 2π = 6π). That's the logic that stands behind the following rule.
Knowing the circumferences, we can find the ratio between revolutions per minute. If the larger wheel rotates at x rmp, and the smaller wheel rotates at y rpm, then the ratio of rpm's is a reciprocal of the ratio of their circumferences:
x/y = 4π/(6π) = 2/3
So for every 2 revolutions that the larger wheel makes, the smaller one revolves 3 times.
Therefore we know the ratio between rpm, but we don't know the actual difference, since we need to know the rotation speed of at least one of the wheels.
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