Hi. I have recently attempted to solve the following problem:
From the top of a 135 foot observation tower, a park ranger sights two forest fires on opposite sides of the tower. If their angles of depression are 42.5 degrees and 32.6 degrees, how far apart are the forest fires?
Here is how I solved this problem. I drew my diagram, with the 135 foot tower, x's marking each forest fire, and lines from the tower to each x, one with the angle being formed labeled as 42.5 degrees, the other as 32.6 degrees. I labeled the distance between Fire #1 and the tower a, and the distance between fire number 2 and the tower b. Then I solved for a and b. To solve for a, I wrote Tangent 32.6 = a/135. I think this is correct because a is opposite the angle, and the tower is adjacent to it. Then I multiplied 135 by the tangent of 32.6 (which is about .64), and got about 86.3 for a. Then I solved for b- Tangent 42.5 = b/135, solved for b the same way, and got 123.7 (Tan 42.5 =.92*135 = 123.7). The distance between the two fires, is a+b, so I added 123.7 to 86.3 (not these rounded figures, the actual answers the calculators gave me), and I got an answer of about 210.
However, the answer key I consulted with later said the correct distance between the fires was 358 feet. Can someone please explain to me where I went wrong, as I was pretty confident with my answer? Thank you so much in advance.