I have a pencil that always rolls around on a slanted surface. One end is wider and heavier than the other. So whenever it rolls, it goes in a wide circle. Otherwise, the pencil is featureless, only becoming steadily wider towards one end.

The difference between the two diameters on the two ends is exactly 144 times smaller than the length of the pencil. If the pencil is pointing uphill on a slanted surface, how many times will it spin until it points downhill?

To keep the diameters integers lets make the length 144.

To keep the arch the smallest lets make the small diameter 0 forcing the large diameter to be 1.

This means the 0 end of the pencil never moves and is the center of the arch.

The length of the arch is 144 pi and one revolution travels 1 pi.

The solution is 144 pi / 1 pi = 144 / 1 = 144.

P.S. Just read the other posts. I can see how the non-tangent measure of length could be more correct, However, I don't think thats what they were looking for.

*Edited on ***February 28, 2004, 10:40 pm**
*Edited on ***February 28, 2004, 10:54 pm**