Question

Find the height of the chimney when it is found that on walking towards it $50$ m in the horizontal line through its base, the angle of elevation of its top changes from ${30}^{\circ }$ and to ${60}^{\circ }$

• A. $25\sqrt{3}$m
• B. $25$ m
• C. $25\sqrt{4}$ m
• D. $\frac{25}{\sqrt{3}}$ m

Answer 1

The correct option is A $25\sqrt{3}$m

Let $PQ=h$ be the height of chimney.

$A$ and $B$ are the two points $50$ m apart.
In $△APQ$, we have
$\tan {30}^{\circ }=\frac{h}{AP}$
$\Rightarrow AP=h\cot {30}^{\circ }=\frac{h}{AP}$
$\Rightarrow AP=h\cot {30}^{\circ }$ .... (i)
and in $△QBP$, we have
$\tan {60}^{\circ }=\frac{h}{BP}$
$\Rightarrow BP=h\cot {60}^{\circ }$ .... (ii)
Since $AP-BP=50$
Thus $h(\cot {30}^{\circ }-\cot {60}^{\circ })=50$
$\Rightarrow h=\frac{50}{(\sqrt{3}-\frac{1}{\sqrt{3}})}=\frac{50\sqrt{3}}{3-1}$
$=\frac{50\sqrt{3}}{2}$

$=25\sqrt{3}$ m , which is the required height