2013 AMC 12B Problem 11

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

Answer: A

Concepts:3D geometry distance formula

Difficulty rating: 1610

Small hint:

Use coordinates with east, north, up as

x, y, z;

the bees only get farther apart over time

Big hint:

Compute the distance right before and right after each foot to find where it crosses

10

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

Answer: A

Difficulty rating: 1610

Small hint:

Big hint:

Solution:

Problem 11 in other years