An airplane is sighted at the same time by two ground observers who are 2 miles apart and both directly west of the airplane. They report the angles of elevation as 11˚ and 20˚. How high is the airplane? Round to the nearest hundredth of a mile.

Answer 1

make your sketch
drop an altitude from the plane to the ground to get 2 right-angled triangles
let the height of the plane be h
let the base of the 11° triangle be x
then the base of the other triangle is 2-x

from the 11° triangle:
tan11 = h/x --->h = xtan11
from the other triangle:
tan20 = h/(2-x) --> h = (2-x)tan20

xtan11 = (2-x)tan20
xtan11 = 2 - xtan20
xtan11 + xtan20 = 2
x(tan11 + tan20) = 2
x = 2/(tan11 + tan20)

h = x tan11
= 2tan11/(tan11 + tan20)

= ...

only now would I go to my calculator to do the evaluation.

Answer 2

an airplane is sighted at the same time by two ground observers who are 4 miles apart and are both directly west of the airplane. They report the angles of elevation as 11 and 22. What is the altitude of the airplane? Round to the nearest tenth

Answer 3

1.67 miles