Question

If two towers of heights h₁, and h₂ subtend angles of 30° and 60° respectively at the midpoint of the line joining their feet, then the ratio of h₁: h₂ is

• A. 2:1
• B. 1:2
• C. 3:1
• D. 1:3

Answer 1

The Correct Option is D

Let $h_1$ and $h_2$ be the heights of the two towers. 

Let the distance between the feet of the towers be $2d$. 

The midpoint of the line joining their feet is at a distance $d$ from each tower. 

The angles subtended at the midpoint are $30^\circ$ and $60^\circ$ respectively. 

Then, we have $$ \tan 30^\circ = \frac{h_1}{d} $$ $$ \tan 60^\circ = \frac{h_2}{d} $$ We want to find the ratio $\frac{h_1}{h_2}$. We have $$ h_1 = d \tan 30^\circ \\ h_2 = d \tan 60^\circ $$ So, $$ \frac{h_1}{h_2} = \frac{d \tan 30^\circ}{d \tan 60^\circ} = \frac{\tan 30^\circ}{\tan 60^\circ} = \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{1}{\sqrt{3} \cdot \sqrt{3}} = \frac{1}{3} $$ 

Therefore, the ratio $h_1:h_2$ is $1:3$.