SOLUTION: Two ocean liners leave from the same port in Puerto Rico at 10:00 a.m. One travels at a bearing of N 46°W at 12 miles per hour, and the other travels at a bearing of S 55°W at 14

Algebra ->  Trigonometry-basics -> SOLUTION: Two ocean liners leave from the same port in Puerto Rico at 10:00 a.m. One travels at a bearing of N 46°W at 12 miles per hour, and the other travels at a bearing of S 55°W at 14      Log On


   

Question 636042: Two ocean liners leave from the same port in Puerto Rico at 10:00 a.m. One travels at a bearing of
N 46°W at 12 miles per hour, and the other travels at a bearing of S 55°W at 14 miles per hour.
Approximate the distance between them at noon the same day. Round answer to two decimal places.
a. 20.58 miles
b. 36.44 miles
c. 25.91 miles
d. 22.71 miles
e. 33.22 miles
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
It would probably help to draw a diagram. Let's say the two liners start from the origin on a graph. The first liner would travel up into the 2nd quadrant. Draw a line segment from the origin up into the 2nd quadrant. Label the angle between the line segment and the y-axis as 46 degrees. Label the length of this segment as 24 (since the line has been traveling for two hours at 12 miles per hour).

The second liner would travel down into the 3rd quadrant. Draw a segment from the origin down into the 3rd quadrant. Label the angle between that segment and the y-axis as 55 degrees. And label the length of this segment as 28 (2 hours at 14 miles per hour).

Draw a line segment that connects the ends of the line segments you've already drawn. The length of this segment is the distance between the ships at noon. This is what we need to find.

Looking at the diagram we should recognize that the three drawn segments form a triangle. And we should know that with the Law of Sines and/or the Law of Cosines we can find any part (side or angle) of a triangle as long as you have one side and any other two parts. Right now we know two sides. We need another part of the triangle.

Looking at the diagram we should notice that the two labeled angles and one of the angles inside the triangle are adjacent angles that make a straight angle. So the three of them must add up to 180 degrees. We can use this to find the angle in the triangle (which will be the third part we need):
180 - (46+55) = 180 - (101) = 79
The angle in the triangle between the 46 and 55 degree angles is 79 degrees.

With the information we have, two sides and an angle, it is not possible to use the Law of Sines to find a missing side. (We could use it to find a missing angle but we are not interested in those.) So we will need to use the Law of Cosines. Since the side we are looking to find is opposite the angle we know, the Law of Cosines sets up this way:
x%5E2+=+%2824%29%5E2%2B%2828%29%5E2-2%2824%29%2828%29cos%2879%29

I'll leave it up to you and your calculator to finish. Just simplify the right side (using the Order of Operations (aka PEMDAS), of course). Then find the square root of the simplified right side.