# 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...

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.

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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)`

Therefore

`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)