A distance between two places, R and K, is 35000 meters. The location of an
airplane is simultaneously found out from these two places. From R side, it
is seen straight over K in the height 15,65 degrees and from K it is seen
straight over R in the distance 28,48 degrees. In what hight over see is
this airplane, if both R and K are in the height 100 meters over see?
Regardless of which place we want to use to solve this problem, we have
only one piece of useful information, the angle. We need more than that.
Fortunately, we can obtain the horizontal distance of the airplane from
the sites, and use that to determine the height.
We'll call the angle at site R angle A, and the angle at site K angle B.
The height of the airplane will be h.
The distance of the plane from R is d. The distance of the plane from
K is (35000 - d).
With that information, we begin the calculations.
h / d = tan A h = (tan A) * d
Also:
h / (35000 - d) = tan B h = (tan B) * (35000 - d)
Obviously, the plane is the same distance above the ground when seen
from either location, so h is the same in both equations. That means
we can set the right side of one equation equal to the right side
of the other equation.
d * tan A = (35000 - d) * tan B
d * tan A = (35000 * tan B) - (d * tan B)
(d * tan A) + (d * tan B) = (35000 * tan B)
d (tan A + tan B) = (35000 * tan B)
d = (35000 * tan B) / (tan A + tan B)
tan A = tan 15.65 = 0.28015
tan B = tan 28.48 = 0.5425
d = (35000 * 0.5425) / (0.28015 + 0.5425) = 18987.5 / 0.82265
= 23080.8971 meters << round up, to 23081 meters
The plane is directly over a point 23081 meters from R,
and 11919 meters from K.
Using the first formula above, for the tangent of the angle
at R, we can now find the plane's height AGL (above ground level).
h = (tan A) * d
h = 0.28015 * 23081 meters = 6466.142 meters
Verify by using the numbers relating to K.
h = (tan B) * (35000 - d)
h = 0.5425 * 11919 meters = 6466.06 meters
The minor difference in the decimal parts is caused by earlier rounding.
The problem asks for the plane's height MSL (mean sea level).
We simply add the 100 meters for the ground's height MSL.
6466 meters + 100 meters = 6566 meters MSL
