JEE Main 2023 — Heights and Distances Question with Solution

Given that from the top $A$ of a vertical wall $AB$ of height $30 \mathrm{m},$ the angles of depression of the top $P$ and bottom $Q$ of a vertical tower $PQ$ are ${15}^{\circ }$ and ${60}^{\circ }$ respectively,

$B$ and $Q$ are on the same horizontal level and$C$ is a point on $AB$ such that $CB\mathit{=}PQ.$

The required diagram will be

From the diagram,

$\Rightarrow \tan 60°=\frac{AB}{BQ}$

$\Rightarrow \frac{30}{x}=\sqrt{3};$

$\Rightarrow x=10\sqrt{3} \mathrm{m}$

Also, $\tan 15°=\frac{AC}{CP}$

$\Rightarrow \frac{30-h}{x}=2-\sqrt{3}$

$\Rightarrow 30-h=10\sqrt{3}\left(2-\sqrt{3}\right)$

$\Rightarrow 30-h=20\sqrt{3}-30$

$\Rightarrow h=60-20\sqrt{3}$

Now area of the quadrilateral $BCPQ$ is $BC\times BQ$

$\therefore$ area$=hx$

$=(60-20\sqrt{3})10\sqrt{3}$

$=200\sqrt{3}\left(3-\sqrt{3}\right)$

$=600\left(\sqrt{3}-1\right) {\mathrm{m}}^2$

Hence, this is the correct option.