Expert: Sherman D. - 12/27/2006

Question
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 looked at your work and saw that it was correct too, but I think you used a different method. 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. By the way, thank you for all of the help you have provided so far.
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The text above is a follow-up to ...

-----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.  I'm one of those students that has to draw a diagram in order to get the picture, so if possible could you help me out? Thanks!
-----Answer-----
When you draw it out, it helps.

y * tan(37) = (y - 250) * tan(50)
tan(37)/tan(50) = (y - 250)/y
tan(37)/tan(50) = 1 - (250/y)
tan(37)/tan(50) - 1 = (-250/y)
y = (-250)/((tan(37)/tan(50)) - 1)

Unless I'm doing this wrong, the ship is 680 from the Tower and 580 from the rocks, and the Tower is 512 feet tall.

To make sure I'm doing this right, i would go to answers.yahoo.com just to make sure.

Answer
Here is how they get it.

sin(50 - 37)/250 = sin(37)/y
sin(13)/250 = sin(37)/y
y = sin(37)/(sin(13)/250)
y = (250sin(37))/(sin(13))

if you draw out one big right triangle, then a little distance closer to the 90 degree angle you make another line extending from that point to the top of the tower, that is your diagram.

I would go to answers.yahoo.com, because its been awhile since i have done these.