1973 AMC 12 Problem 20

Question

A cowboy is

4

miles south of a stream which flows due east. He is also

8

miles west and

7

miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is

4+\sqrt{185}

16

17

18

\sqrt{32}+\sqrt{137}

Answer: C

Concepts:reflection (geometry)distance formula optimization

Difficulty rating: 1850

Small hint:

Reflect the cowboy’s starting point across the straight stream

Big hint:

After reflection, the two-leg trip through the stream becomes one straight segment to the cabin

Solution:

Take the stream as the

x

-axis and put the cowboy at

C=(0,-4).

His cabin is then

H=(8,-11).

Reflect

C

across the stream to

D=(0,4).

For any point

S

on the stream,

CS=DS,

so minimizing

CS+SH

is the same as minimizing

DS+SH.

This occurs when

D,

S,

H

are collinear. The minimum distance is

DH=\sqrt{8^2+15^2}=\sqrt{289}=17.

Therefore, the correct answer is C.