Solve Angle of Depression Problem for Office Building Distance

Question

Office Buildings: The angle of depression from the top of a 320 foot office building to the top of a 200 foot office building is 55°. How far apart are the buildings?

Accepted Answer

84.0 feet

Answer

Step 1: Calculate the vertical difference between the buildings: $320 - 200 = 120$ feet. This is the opposite side of the right triangle formed by the horizontal distance (x, adjacent side) and the line of sight. Step 2: Recognize that the angle of depression (55°) is equal to the angle of elevation from the shorter building to the taller building (alternate interior angles). Step 3: Use the tangent function: $\tan(55°) = \frac{120}{x}$. Step 4: Rearrange to solve for x: $x = \frac{120}{\tan(55°)} \approx \frac{120}{1.4281} \approx 84.0$ feet.