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Expert: - 10/18/2010


Question
From an 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 tower are in the same vertical plane and the distance from A to B is 1.1km. Determine the height of the tower.
Could u show me step by step how to answer this question. Dont know why but it has me stumped. I need this last answer to finish my math, then i can write a test ..

Answer
I think I answered this already, but here's another approach.

The distance from A to B is 1.1 km.  Say A is x km from the tower, then B is 1.1-x km from the tower.  Call the height of the tower h.
Let C be the tangent of the angle at point A and D be tangent of the angle at point B.

This makes C = h/x and D = h/(1.1-x).
It can be seen that from h/x = C, h = xC and from
h/(1.1-x) = D, h = (1.1-x)D.

Since we have h = xC and h = (1.1-x)D, we know that xC = (1.1-x)D.
Multiplying the equation out gives xC = 1.1D - xD.

Adding xD to both sides gives xC + xD = 1.1D.

Factoring out the x on the left gives x(C+D) = 1.1D.

Dividing both sides by (C+D) gives x = 1.1D/(C+D).

Put in C = tan(A) = tan(6°) and D = tan(B) = tan(13°).

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