SOLUTION: An airplane is sighted at the same time by two ground observers who are 19.4 miles apart and both directly west of the airplane. They report the angles of elevation as 12° and 34Â

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Question 1199063: An airplane is sighted at the same time by two ground observers who are 19.4 miles apart and both directly west of the airplane. They report the angles of elevation as 12° and 34° . How high is the airplane?
Found 2 solutions by math_tutor2020, MathTherapy:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

This is one way to draw out the diagram

A & B = observation points
C = point on the ground directly under the airplane
D = the airplane's location in the air

x = length of segment BC
h = airplane's height = length of segment DC

Let
u = tan(34)
v = tan(12)
as these items will be helpful later.

Focus on right triangle BCD
tan(angle) = opposite/adjacent
tan(B) = DC/BC
tan(34) = h/x
u = h/x
h = ux
We'll use this as a substitution in the next section below.

Now focus on right triangle ACD.
tan(angle) = opposite/adjacent
tan(A) = DC/AC
tan(A) = DC/(AB+BC)
tan(12) = h/(19.4+x)
v = h/(19.4+x)
v(19.4+x) = h
19.4v+vx = h
h = 19.4v+vx
ux = 19.4v+vx ....... plug in h = ux
ux-vx = 19.4v
x(u-v) = 19.4v
x = 19.4v/(u-v)

So,
h = ux
h = u*19.4v/(u-v)
h = 19.4uv/(u-v)
We have a handy formula to calculate the height.
It connects together the tangents of the angles of elevation, and the distance between the observation points.

The more generalized formula is
h = d*u*v/(u-v)
where d is the distance between observation stations
u,v are the tangents of the angles of elevation, with u > v
Since u > v, this means the larger angle is associated with variable 'u'.

Let's wrap things up
h = 19.4uv/(u-v)
h = 19.4*tan(34)*tan(12)/(tan(34)-tan(12))
h = 6.0209756995424
h = 6.02
Make sure your calculator is in degree mode.


Answer: Approximately 6.02 miles
Round your answer however your teacher instructs.
To convert to feet, multiply by 5280
6.02 miles = 5280*6.02 = 31,785.6 feet

Similar question:
https://www.algebra.com/algebra/homework/Vectors/Vectors.faq.question.1198061.html
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
An airplane is sighted at the same time by two ground observers who are 19.4 miles apart and both directly west of the airplane. They report the angles of elevation as 12° and 34° . How high is the airplane?
.
Let height of airplane, or AD be h
With ∡ACD being 34o, ∡ACB = 180 - 34 = 146o. 
As ∡ACD are 34o and 146o, respectively, ∡BAC = 180 - (12 + 146) = 180 - 158 = 22o.

Using the Law of Sines, we get: 
                           AB * sin (22o) = 19.4 * sin (146o) ---- Cross-multiplying
                                      

              Now we can say that: 
                                 
            Height of airplane, or

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