Speeding

Trigonometry Precalc/Calc Word Problem…?

Milwaukee, Wisconsin and Grand Haven, Michigan are directly across from each other over Lake Michigan. One night, a law enforcement boat leaves Milwaukee on a coarse of 105° at the same time a small smuggling craft leaves Grand Haven at an angle of 195°. The law enforcement boat averages 23 knots and collides with the smuggling craft–what was the smuggling craft’s average speed?

First off, it would be nice if you could let us know that Milwaukee is due west of Grand Haven. I had to look it up (not all of us are familiar with the geography of that part of the world).

http://www.salmonuniversity.com/Great%20Lakes/Lake%20Michigan%20Map.gif

Now we can draw triangle ABC, where
A = Milwaukee
B = Grand Haven
C = positing where boats collide

Since law enforcement boat leaves Milwaukee on a course of 105° (east of north I assume)
∠CAB = 15°
Since the smugglers’ boat leave Grand Haven at an angle of 195°
∠CBA = 75°
and
∠ACB = 180 – 15 – 75 = 90°

Let x = speed of smugglers’ boat
tan(15°) = BC/AC = x/23
x = 23 tan(15°)
x = 6.1628

Daniel Johnston – Speeding Motorcycle – Madison, WI