Question

The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?

$2$ o' clock

$3$ o' clock

$4$ o' clock

$6$ o' clock

$8$ o' clock

Solution:

The disk of radius $10$ rolls externally around the clock face of radius $20$ . If the point of tangency moves through central angle $\theta$ around the clock, the disk rotates through $\frac{20+10}{10}\theta=3\theta$ relative to its original direction.

The arrow next points upward when $3\theta$ is a positive multiple of $2\pi$ . The first time this happens is $\theta=\frac{2\pi}{3}$ , which is one-third of the way around the clock from 12 o'clock, namely 4 o'clock.

Thus, C is the correct answer.

2015 AMC 10A Problem 14

Solution:

Problem 14 in Other Years