Question

Two towers of heights $h_1$ and $h_2$ subtend angles ${60}^0$ and ${30}^0$ respectively at the midpoint of the line joining their feet. $h_1:h_2=$

• A. $1:2$
• B. $1:3$
• C. $2:1$
• D. $3:1$

Answer 1

The correct option is D $3:1$

Given that,

Two towers have height $h_1$ and $h_2$ respectively.

They subtend ${60}^o$ and ${30}^o$ at the midpoint of line joining their feet.

Based upon the given information, we can draw the above figure.

Here, $AB=h_{1}andDE=h_2$
$BC=CD=d$

We know that, $\tan \theta =\frac{\text{Opposite Side}}{\text{Adjacent Side}}$

Hence, in $△CDE,\tan {30}^o=\frac{h_2}{d}....\left(i\right)$

and in $△ABC,\tan {60}^o=\frac{h_1}{d}....\left(ii\right)$

Divinding $\left(i\right)$ and $\left(ii\right)$, we get:

$\frac{\tan {30}^o}{\tan {60}^o}=\frac{\frac{h_2}{d}}{\frac{h_1}{d}}$

$\Rightarrow \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}}=\frac{h_2}{h_1}$ $[\because \tan {30}^o=\frac{1}{\sqrt{3}}and\tan {60}^o=\sqrt{3}]$

$\Rightarrow \frac{1}{3}=\frac{h_2}{h_1}$

$\therefore \frac{h_1}{h_2}=\frac{3}{1}$

Hence, $h_1:h_2=3:1$.