A plane flies a straight course. On the ground directly below the flight path, observers 3 miles apart spot the plane at the same time. The plane's angle of elevation is 69° from one observation point and 40° from the other. How high is the plane?

Picture for reference below.

Question

A plane flies a straight course. On the ground directly below the flight path, observers 3 miles apart spot the plane at the same time. The plane's angle of elevation is 69° from one observation point and 40° from the other. How high is the plane? 

Picture for reference below.

anonymous · Answer

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anonymous · Answer

start by using the law of sines to determine the distance between the plane and the observers

dumbcow · Answer

that is one way...a simpler way (though it involves more algebra) is to use tangents

tan(40) = h/x
tan(69) = h/(3-x)

solve top equation for x and substitute it into bottom equation
then solve for h

anonymous · Answer

Alright, I just looked up law of sins.  We learn that tomorrow, but it seems pretty simple.  I have 3/sin(71) = b/sin(69) so far.  I guess the obvious move is to solve for b here.

dumbcow, I'll try that right now.

anonymous · Answer

dumbcow, I feel pretty dumb right now, but I'm havin' trouble doing your method.

dumbcow · Answer

thats ok...try the other way, use law of sines as above, solve for b then use sin ratio to get height
that probably is better way for students learning the material

just so you know, the solution using tangents is
$$h = \frac{3\tan 69 \tan 40}{\tan 69 + \tan 40}$$