solve this---**Diagram Description:**
The diagram shows two circles in contact. The smaller circle on the left is labeled with its center as point A. The larger circle on the right is labeled with its center as point B. An curved arrow indicates a counter-clockwise rotation around circle A. Circle A is shown tangent to circle B.
**Question:**
17. In the figure above, the radius of circle A is 1/3 the radius of circle B. Starting from the position shown in the figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A first reach its starting point?
**Options:**
(A) 3/2
(B) 3
(C) 6
(D) 9/2
(E) 9
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In this problem, we have two circles. Circle A has a radius that is one-third the radius of circle B. Circle A rolls around the outside of circle B. We need to determine after how many revolutions of circle A will its center first return to the starting point. Let's analyze this step by step.
To understand this problem, let's examine what happens when circle A rolls around circle B. The center of circle A follows a circular path with radius equal to the sum of both circles' radii, which is r_A plus r_B. The circumference of this path is 2π times r_A plus r_B. When the center of A completes one full revolution around B, it travels exactly this distance. As circle A rolls, it also rotates around its own center.
Now let's calculate how many revolutions circle A makes before its center returns to the starting point. The center of A travels a distance equal to the circumference of the path circle, which is 2π times r_A plus r_B. For each revolution of circle A around its own center, it travels a distance equal to its own circumference, which is 2π times r_A. So the number of revolutions is the ratio of these distances. Since r_A equals one-third of r_B, we can substitute and find that circle A makes 4 total revolutions. However, this includes both the 1 revolution around circle B and 3 revolutions relative to the path.