Question

a plane flies a straight course. On the ground directly below the flight path. observers 2 miles apart spot the plane at the same time. The planes angle of elevation in 46 degrees from on observation point and 71 degrees from the other. How high is the plane?

Answer 1

Kindly refer to the attached figure for reference

Let A and B be the two points where we have the two observers.  Let C be the point above at which the plane flies

Angle A = 46 degrees and Angle B = 71 degrees

AB = 2 miles

Let AD = x , then DB = 2-x

Let the vertical distance between the plance and ground represented by CD be y

As per trigonometric functions, tan θ = opposite side/adjacent side

tan 46 = y/x

tan 71 = y/(2-x)

tan 46  = 1.03553

tan 71 = 2.90421

1.03553 = y/x

y = 1.03553x

2.90421 = y/(2-x)

2.90421*(2-x) = y

5.80842 - 2.90421x = y

5.80842 - 2.90421x  = 1.03553x

5.80842 = 3.93974x

x = 1.4743 miles

y=1.03553x

y = 1.4743 * 1.03553

y =1.5267 miles

CD = 1.5267 miles

The plane is 1.5267 miles from the ground