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 height of the tower be $h$. Using the tangent function in $\triangle ABD$: $$\tan 45^\circ = \frac{AB}{BD} \Rightarrow BD = 50 { m}$$ Now, in $\triangle ACC'$: $$\tan 30^\circ = \frac{AC'}{C'C}$$ $$\frac{1}{\sqrt{3}} = \frac{50 - h}{50}$$ Solving for $h$: $$50 = 50\sqrt{3} - h\sqrt{3}$$ $$h\sqrt{3} = 50(\sqrt{3} - 1)$$ $$h = 50 \left( 1 - \frac{\sqrt{3}}{3} \right) { m}$$