Question

From the top of a cliff 50 m high, the angles of depression of the top and bottom of a tower are observed to be \(30^\circ\) and \(45^\circ\). The height of the tower is:

• A. \( 50\sqrt{3} \) m
• B. \( 50(\sqrt{3} - 1) \) m
• C. \( 50\left( 1 - \frac{\sqrt{3}}{3} \right) \) m
• D. 50(1 - √3/3) m

Hint:

Trigonometry is useful in solving real-world height and distance problems.

Answer 1

The Correct Option is D

Let the tower's height be \( h \). In \( \triangle ABD \), using the tangent function: \[ \tan 45^\circ = \frac{AB}{BD} \Rightarrow BD = 50 { m} \] In \( \triangle ACC' \), we have: \[ \tan 30^\circ = \frac{AC'}{C'C} \] \[ \frac{1}{\sqrt{3}} = \frac{50 - h}{50} \] Solving for \( h \): \[ 50 = 50\sqrt{3} - h\sqrt{3} \] \[ h\sqrt{3} = 50(\sqrt{3} - 1) \] \[ h = 50 \left( 1 - \frac{\sqrt{3}}{3} \right) { m} \]