Question

From the top of a tower 60 meters high, the angles of depression of top and bottom of a building (house) are 45° and 60° respectively. Find the height of the building and its distance from the tower.

Hint:

To solve problems involving angles of depression or elevation, use trigonometric functions such as tangent, sine, and cosine.

Answer 1

Let: - \( h \) be the height of the building, - \( x \) be the distance of the building from the tower. We will use the tangent function to solve the problem. Since the angles of depression are given, the angles of elevation from the base of the tower to the top and bottom of the building are also \( 45^\circ \) and \( 60^\circ \), respectively, by the alternate angle theorem.
Step 1: For the top of the building:
\[ \tan(45^\circ) = \frac{\text{height of the tower}}{\text{distance from the tower}} = \frac{60}{x} \] \[ \tan(45^\circ) = 1 \Rightarrow 1 = \frac{60}{x} \Rightarrow x = 60 \text{ meters} \]
Step 2: For the bottom of the building:
\[ \tan(60^\circ) = \frac{60 - h}{x} \] \[ \tan(60^\circ) = \sqrt{3} \Rightarrow \sqrt{3} = \frac{60 - h}{60} \] \[ 60 - h = 60\sqrt{3} \Rightarrow h = 60 - 60\sqrt{3} \] \[ h \approx 60 - 103.92 = -43.92 \, \text{meters}. \] Thus, the height of the building is approximately \( 43.92 \, \text{meters} \). The distance of the building from the tower is 60 meters.