Question

Two observers who are 5 km apart simultaneously sight a small airplane flying between them. One observer measures a 51.0-degree inclination angle, while the other observer measures a 40.5-degree inclination angle. At what altitude is the
airplane flying?

Answer 1

Draw a triangle ABC , A and B the position of the observers and C the position of the plane.
Drop an altitude from C to AB, call it CD.
Label angle A as 51 and angle B as 40.5 degrees
Label AD as x, then DB = 5-x

In triangle ACD, cot51 = x/h
x = hcot51
In triangle DBC cot 40.5 = (5-x)/h
5-x = hcot40.5 or
x = 5 - hcot40.5

then hcot51 = 5-hcot40.5
hcot51 + hcot40.5 = 5
h = 5/(cot51 + cot40.5)
= .....