Question

17. Construction: Two office buildings are 51 m apart. The height of the taller building is 207 m. The angle of depression from the top of the taller building to the top of the shorter building is 15 degrees. Find the height of the shorter building to the nearest meter.

Answer 1

The distance and angle of depression form a right triangle.

The horizontal side of the triangle is the 51 meter distance between them. It goes from the top of the shorter building straight across the the same height on the taller building.

The height of the triangle is the difference in heights of the buildings.

Since the angle of depression is 15 degrees the top angle of the triangle is 90 - 15 degrees or 75 degrees. You can set its tangent (opposite over adjacent sides) equal to the 51 meter (opposite side) over the difference in heights (adjacent side).

tan(75) = 51 / d

Multiply both sides by d and divide both sides by tan(75) to express d in terms of the distance and tangent:

The tangent of 75 degrees is 3.73 (rounded).

d = 51 / tan(75) = 51 / 3.73 = 13.7 meters difference in height


Subtract 13.7 meters from the taller building's height of 207 meters:

207 - 13.7 = 193.3 meters which rounds to 193 meters

The shorter building is 193 meters tall.