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?
<pre>*[illustration ADC_1199063.png].
Let height of airplane, or AD be h
With &#8737ACD being 34<sup>o</sup>, &#8737ACB = 180 - 34 = 146<sup>o</sup>. 
As &#8737ACD are 34<sup>o</sup> and 146<sup>o</sup>, respectively, &#8737BAC = 180 - (12 + 146) = 180 - 158 = 22<sup>o</sup>.

Using the Law of Sines, we get: {{{matrix(2,3, sin (BAC)/BC, "=", sin (ACB)/AB, sin (22^o)/19.4, "=", sin (146^o)/AB)}}}
                           AB * sin (22<sup>o</sup>) = 19.4 * sin (146<sup>o</sup>) ---- Cross-multiplying
                                      {{{matrix(1,9, AB, "=", (19.4 * sin(146^o))/sin (22^o), "=", "28.95929361,", or, 29, miles, "(approximately)")}}}

              Now we can say that: {{{matrix(1,5, sin (B), "=", O/H, "=", AD/AB)}}}
                                {{{matrix(1,3, sin (12^o), "=", h/29)}}} 
            <font color = red><font size = 4><b>Height of airplane</font></font></b>, or {{{highlight_green(matrix(1,8, h, "=", 29 * sin (12^o), "=", "6.029439034,", or, highlight(matrix(1,2, 6.03, miles)), "(approximately)"))}}}</pre>