From and observation tower that overlooks a runway, the angles of depression of point A, on one side of the runway, and point B, on the opposite side of the runway are , 6 degrees and 13 degrees respectively. The points and the towers are in the same vertical plane and the distance from A to B is 1.1 km. Determine the height of the tower.
The figure is below attached. The values of all angles are given in the figure.
From the theorem of sinuses in triangle `ABD` one can write
`(AB)/sin(7) =(BD)/sin(6) rArr BD = AB*sin(6)/sin(7)`
From the theorem of sinuses in triangle `BCD` one can write
`(CD)/sin(13) =(BD)/sin(90) rArr CD =BD*sin(13)`
`CD = AB*sin(6)/sin(7)*sin(13)`
`CD = 1100*sin(6)/sin(7)*sin(13) =212.23 m`
Answer: the height of the tower is 212.23 m
on the 6degree side let x be the distance of point A from the foot of the tower and the distance of the point B from the foot of the tower will be 1.1 - x ( on the 13 degree side).
Let h be the height of the tower.
x/h = tan6 and (1.1 - x)/h = tan13.
x=h tan6 and 1.1 - x = h tan13 = 1.1 - h tan6
so, h tan13 + h tan6 = 1.1
hence h(tan6 + tan13) = 1.1
h = 1.1 / (tan6+tan13)