Question

The angles of elevation of top of a pole from two points $A$and $B$ on the horizontal line lying on opposite side of the pole are observed to be ${30}^{\circ }$ and ${60}^{\circ }$ If $AB=100m$, then height of the pole is

• A. $20\sqrt{3}m$
• B. $15\sqrt{3}m$
• C. $10\sqrt{3}m$
• D. $25\sqrt{3}m$

Answer 1

Answer: (D)

$25\sqrt{3}m$

Let the height of pole be $PQ=h$ m.

And $QB=d$ be the distance between $B$ and foot of pole.

$\tan {60}^{\circ }=\frac{h}{d}$
$\sqrt{3}=\frac{h}{d}$
$\Rightarrow d=\frac{h}{\sqrt{3}}$....(1)

$\tan {30}^{\circ }=\frac{h}{100-d}$

$\frac{1}{\sqrt{3}}=\frac{h}{100-d}$
$\Rightarrow 100-d=\sqrt{3}h$
Substitute the value of d from the equation (1), we get

$\Rightarrow 100-\frac{h}{\sqrt{3}}=\sqrt{3}h$

$\Rightarrow 100=h(\sqrt{3}+\frac{1}{\sqrt{3}})$
$\Rightarrow 100=h\left(\frac{3+1}{\sqrt{3}}\right)$
By doing cross multiplication
$\Rightarrow 100\sqrt{3}=4h$

$\Rightarrow 4h=100\sqrt{3}$

$\Rightarrow h=25\sqrt{3}m$