AIME II Problem 11 2007

[zefuri]

Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius 72 and rolls along the surface toward the smaller tube, which has radius 24. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance x from where it starts. The distance x can be expressed in the form [unparseable or potentially dangerous latex formula], where a, b, and c are integers and c is not divisible by the square of any prime. Find a+b+c.

[collegebookworm]

The image for this solution is located at


The circumference is [unparseable or potentially dangerous latex formula], meaning that a randomly chosen point on the cylinder’s circumference will have traveled about the surface before coming to rest [unparseable or potentially dangerous latex formula] units later according to its reference. Let’s assume that the large cylinder starts tangent to the circle and we wish to count the distance traveled on the circumference up to the [unparseable or potentially dangerous latex formula] units. As shown in the picture, the tangency point creates a 30-60-90 triangle which means the point of tangency of the two cylinders occurs at the larger cylinder [unparseable or potentially dangerous latex formula] from the bottom and for the smaller, [unparseable or potentially dangerous latex formula] from the bottom. Notice that when the larger cylinder begins to move up the smaller cylinder, there is a smaller radian change for the larger cylinder as for the smaller cylinder; precisely, for every radian traveled on the smaller, the larger cylinder will have turned [unparseable or potentially dangerous latex formula] radians. As the larger cylinder will only travel on the smaller one from one point of tangency to the other, we must find the circumference traveled by the larger cylinder. We must also consider the fact that when we begin changing surfaces, the point of reference “shiftsâ€