2013 AMC 12B Problem 11

Problem 11 of 25IntermediateGeometry

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?

Show solution

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.