Q39 / 90 · JEE Main 2019 (10 Jan Shift 1) · Mathematics

Consider a triangular plot $A B C$ with sides $A B = 7 m,$ $B C = 5 m$ and $C A = 6 m .$ A vertical lamp-post at the mid-point $D$ of $A C$ subtends an angle $30 °$ at $B .$ The height (in $m$ ) of the lamp-post is:

Solution

The length of the median $B D$ is given by

$B D = \frac{1}{2} \sqrt{2 (B C^{2} + B A^{2}) - {(A C)}^{2}}$

$\Rightarrow B D = \frac{1}{2} \sqrt{2 (25 + 49) - 36}$

$\Rightarrow B D = \frac{1}{2} \sqrt{112}$

$\Rightarrow B D = \frac{4 \sqrt{7}}{2} = 2 \sqrt{7} m .$

Now, let $D T$ be the tower of height $h m$ and it subtends $∠ T B D = 30 °$ at the vertex $B .$

Then, in $∆ T D B$ we have $\tan (∠ T B D) = \frac{T D}{B D}$

$\Rightarrow \tan 30 ° = \frac{h}{2 \sqrt{7}}$

$\Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{2 \sqrt{7}}$

$\Rightarrow h = 2 \sqrt{\frac{7}{3}}$

$\Rightarrow h = \frac{2}{3} \sqrt{21} m .$

Hence, the correct answer is option: 'B'.