Question

Two pillars of equal heights stand on either side of a road which is 100 m wide. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are ${60}^{\circ }$ and ${30}^{\circ }$. Find the height of each pillar and position of the point on the road. [Take $\sqrt{3}$=1.732] [3 MARKS]

Answer 1

Concept: 1 Mark
Application: 2 Marks

Let AB and CD be two pillars, each of height h metres and let AC be the road such that AC = 100 m.

Let O be the point of observation on AC.

Let OA = x metres and OC = (100 - x) m

Also, $\angle AOB={60}^{\circ }$ and $\angle COD={30}^{\circ }$

$AB\perp AC$ and $CD\perp AC$

From right $\Delta OAB$, we have



$\frac{AB}{OA}=tan{60}^{\circ }=\sqrt{3}$

$\Rightarrow \frac{h}{x}=\sqrt{3}\Rightarrow h=\sqrt{3}x$

From right $\Delta OCD$, we have

$\frac{CD}{OC}=tan{30}^{\circ }=\frac{1}{\sqrt{3}}$

$\Rightarrow \frac{h}{(100-x)}=\frac{1}{\sqrt{3}}\Rightarrow h=\frac{(100-x)}{\sqrt{3}}$ .....(ii)

$\sqrt{3}x=\frac{(100-x)}{\sqrt{3}}\Rightarrow 3x=(100-x)$

$\Rightarrow 4x=100\Rightarrow x=25$

Putting x = 25 m in (i), we get

$h=(25\times \sqrt{3})=(25\times 1.732)=43.3$

Hence, the height of each pillar is 43.3 m and the point of observation is 25 m away from the first pillar.