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June 23rd, 2017, 10:32 PM  #1 
Senior Member Joined: Apr 2008 Posts: 192 Thanks: 3  How do you solve this bearing problem?
From the top of a 50m observation tower, a fire ranger observes smoke in two locations. One is on a bearing of 40 degrees with an angle of depression of 8 degrees and the other is on a bearing of 205 degrees with an angle of depression of 13 degrees. How far apart are the sources of smoke? I am very confused about how to incorporate bearings into trigonometry. Please help me. Thanks a lot. 
June 23rd, 2017, 11:22 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,860 Thanks: 966 
can you assume that the fires both occur at altitude 0m?

June 24th, 2017, 05:00 AM  #3 
Math Team Joined: Jul 2011 From: Texas Posts: 2,737 Thanks: 1387 
Assuming both fires have the same vertical elevation as the base of the tower ... The horizontal distance from the base of the tower to the first fire is $d_1=\dfrac{50}{\tan(8^\circ)}$ The horizontal distance from the base of the tower to the second fire is $d_2=\dfrac{50}{\tan(13^\circ)}$ the angle between their respective bearings is $205^\circ  40^\circ =165^\circ$ distance between the two fires, $D = \sqrt{d_1^2+d_2^2 2d_1d_2\cos(165^\circ)}$ 
July 2nd, 2017, 12:08 AM  #4 
Senior Member Joined: Apr 2008 Posts: 192 Thanks: 3 
Thanks a lot, skeeter.


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