Problem:
Two bees start at the same spot and fly at the same rate in the following directions. Bee travels 1 foot north, then 1 foot east, then 1 foot upwards, and then continues to repeat this pattern. Bee travels 1 foot south, then 1 foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly 10 feet away from each other?
Answer Choices:
A. $$A$ east, west$
B. $$A$ north, south$
C. $$A$ north, west$
D. $$A$ up, south$
E. $$A$ up, west$
Solution:
Suppose that the two bees start at the origin and that the positive directions of the , and coordinate axes correspond to the directions east, north, and up, respectively. Note that the bees are always getting farther apart from each other. After bee has traveled 7 feet it will have gone 3 feet north, 2 feet east, and 2 feet up. Its position would be the point . In the same time bee will have gone 4 feet south and 3 feet west, and its position would be the point . This puts them at a distance
After this moment, bee A will travel east to the point and bee will travel west to the point . Their distance after traveling one foot will be
Hence bee is traveling east and bee is traveling west when they are exactly 10 feet away from each other.
The problems on this page are the property of the MAA's American Mathematics Competitions