### Trig problem - angle of depression - solution

The problem was asking us to find AD and CD, AD being the distance between the airplane and the base of the control tower, and CD being the height of the control tower of the airport. We are given the angles of depression.

First, let's find AD. That is easy. In the right triangle ABD, we know one of the legs (AB) and the angle A (33°). So, we can use COSINE.

cos 33° = 8.5 km / AD

from which AD = 8.5 km / cos 33° = 10.135 km

Then, we go on to find CD. For that, we need to calculate the two lengths BD and BC and subtract those.

For both BD and BC we can use the TANGENT function, but from two different triangles.

From triangle ABC we get

tan 30° = BC /8.5 km

from which BC = 8.5 km (tan 30°) = 4.907477 km

And from triangle ABD we get

tan 33° = BD /8.5 km

from which BD = 8.5 km (tan 33°) = 5.519964542 km.

Subtracting the two, we get BD - BC = 0.612 km or 612 meters.

Oh my, that is a high control tower! According to Wikipedia, the world's tallest free standing control tower is the 132.2 m (434 feet) Suvarnabhumi Airport control tower, 25 km east of Bangkok, Thailand. So this one is over three times that high. Shows you how easily math word problems can be "out of this world."