Question

Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ 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?

$A$ east, $B$ west

$A$ north, $B$ south

$A$ north, $B$ west

$A$ up, $B$ south

$A$ up, $B$ west

Solution:

Take east, north, up as $x, y, z.$ After $7$ feet bee $A$ is at $(2, 3, 2)$ and bee $B$ is at $(-3, -4, 0),$ a distance $\sqrt{78} \lt 10.$ On the next foot bee $A$ moves east to $(3, 3, 2)$ and bee $B$ moves west to $(-4, -4, 0),$ a distance $\sqrt{102} \gt 10.$ So they pass through $10$ feet apart while $A$ heads east and $B$ heads west. Thus, the correct answer is A.

2013 AMC 12B Problem 11

Solution:

Problem 11 in Other Years