Question

A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is

and that the angle of depression to the bottom of the tower is
. How tall is the tower?

Answer 1

461.94 feet

Explanation:

Let's denote the height of the tower as h. Then we can set up two right triangles as follows:

Triangle 1: the person, the top of the tower, and the point where the person is standing

Triangle 2: the person, the bottom of the tower, and the point where the person is standing

In triangle 1, we have the opposite side (height of the tower) and the adjacent side (distance from the building to the tower), and we know the tangent of the angle of elevation:

tan(30°) = h/400

h = 400 * tan(30°)

h ≈ 231.05 feet

In triangle 2, we have the opposite side (height of the tower) and the adjacent side (distance from the building to the tower), and we know the tangent of the angle of depression:

tan(60°) = h/400

h = 400 * tan(60°)

h ≈ 692.82 feet

Since the height of the tower must be the same in both triangles, we can take the average of these two values:

(h1 + h2) / 2 = (231.05 + 692.82) / 2

h ≈ 461.94 feet

Therefore, the height of the tower is approximately 461.94 feet.