Find the angles between their courses.

Question

Pre-calculus

 

A cruise ship and a freight train leave port at the same time and travel straight line courses at 30km/hr, respectively.Two hours later they are 50 km apart. Find the angles between their courses.

Answer 1

There is a triangle formed by the port, and the positions of the two ships after 2 hours.

 

Two of the sides are equal -- the sides from the port to the ships' positions.  Since they have traveled 2 hours at 30 km/hr, those two sides are 60 km.  The other side, that connects the positions of the two ships, is 50 km.

 

When you know all three sides of a triangle, you use the law of cosines to find any of the angles.

 

The law says:

 

c² = a² + b² - 2ab cos C

 

You're trying to find the angle between the ships, so that's C.  The side opposite that angle is c, and that's the side connecting the two ships -- 50 km

 

a and b are the sides adjacent to the angle you're trying to find, so those are the sides from the port to the ships -- 60 km

 

So...

 

50² = 60² + 60² - 2*60*60 cos C

2500 = 3600 + 3600 - 7200 cos C

2500 = 7200 - 7200 cos C

2500 - 7200 = -7200 cos C

-4700 = -7200 cos C

-4700 / -7200 = cos C

0.65278 = cos C

49.25° = C

 

The angle between their courses s 49.25°.