Question

Two pillars are of equal height on either sides of a road which is 100 m wide. The angles of elevation of the top of the pillars are ${60}^{\circ }$ and ${30}^{\circ }$ at a point on the road between the pillars. Find the position of the point between the pillars and height of each pillar

• A. $43.3m$
• B. $28m$
• C. $45.3m$
• D. $54m$

Answer 1

The correct option is A $43.3m$

Let AB and ED be two pillars each of height h metres Let C be a point on he road BD such that
$BC=x$ metres Then $CD=(100-x)$ metres Given $\angle ACB={60}^{\circ }$ and $\angle ECD={30}^{\circ }$
In $\Delta ABC,\tan {60}^{\circ }=\frac{AB}{BC}$ $\Rightarrow \sqrt{3}=\frac{h}{x}\Rightarrow h=\sqrt{3x}..........\left(i\right)$
In $\Delta ECD,\tan {30}^{\circ }=\frac{ED}{CD}$ $\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{100-x}\Rightarrow h\sqrt{3}=100-x...........\left(ii\right)$
$\therefore$ Subst. the value of h from (i) in (ii) we get
$\sqrt{3x}\times \sqrt{3}=100-x\Rightarrow 3x=100-x\Rightarrow 4x=100\Rightarrow x=25m$
$\therefore$ $h=(\sqrt{3}\times 25)=25\times 1.732m=43.3m$
$\therefore$ The required point is at a distance of $25$ m from the pillar B and the height of each pillar is $43.3m$