Expert: Richard J. Raridon - 12/27/2006

Question
A ship captain at sea uses a sextant to sight an angle of elevation of 37° to the top of a lighthouse. After the ship travels 250 feet directly toward the lighthouse. another sighting is made, and the new angle of elevation is 50°. The ships's charts show that there are dangerous rocks 100 feet from the base of the lighthouse. Find, to the nearest foot, how close to the rocks the ship is at the same time of the second sighting. I've seen the answer to this question but I don't know how the answer was derived. This question was on a Math B regent exam in Jan. 2003.  ACCORDING TO THEIR ANSWER KEY THE CORRECT ANSWER IS 330 FT. It states that I should have used the Law of Sine: sin(13)/250 = sin(37)/y and then calculate cos(50) = x/668.8288536. Once I found the answer it says I was supposed to subtract 100. I read this and was unable to figure out how these numbers came to be.  I would greatly appreciate it if you can send me a diagram according to the answers from the key (method above)  so I get a better understanding.  I understand the Law of sine, but I don't understand where the sin(13) came from. I hope you can help tons. Please and thanx.

Answer
Here's how I would work it.  Let x = distance from the base of the lighthouse at the second sighting and h = height of lighthouse.  Draw the picture and you will have two triangles.  One triangle will have angles of 90, 50 and 40 degrees.  The other triangle will have angles of 130, 37 and 13 degrees.  
Then tan(50) = h/x and tan(37) = h/(x+250).  Solving for x gives you 430 feet so the distance from the rocks is 330 feet.
You can use the law of sines as they did but I think my method is quicker.