Question

Atop 50m tower, Jack sees smoke in two areas. One is on a bearing of 40degrees with an angle of depression of 8degrees, and the other on a bearing of 205degrees with and angle of depression of 13degrees. How far apart are the smoke sources?

I'm not sure how bearings work in this question.

Answer 1

I think we must assume that the tower is in relatively flat plane.
I don't know how well you can sketch 3-D diagrams, but mine has two right-angled triangles.
One showing the first fire, has a base angle of 8° and a height of 50, so the distance of the fire from is:
50/base = tan8, base = 50/tan8°
similarly the 2nd right angled triangle has a base angle of 13° and the distance of the fire from the tower is 50/tan13°
Now the bearing:
the 2nd is 204° and the 1st is 40°
so in now have a triangle with sides 50/tan8 and 50/tan13 with an angle of 204-40 or 165° between them
let the distance between the two fires be x

x^2 = (50/tan8)^2 + (50/tan13)^2 - 2(50/tan8)(50/tan13)cos 165°

I will let you do the button-pushing.