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 $$\displaystyle 60^{\circ}$$ and $$\displaystyle 30^{\circ}$$ 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

Solution

## The correct option is A $$43.3 m$$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 $$\displaystyle \angle ACB=60^{\circ}$$ and $$\displaystyle \angle ECD=30^{\circ}$$In $$\displaystyle \Delta ABC, \tan 60^{\circ}=\frac{AB}{BC}$$ $$\displaystyle \Rightarrow \sqrt{3}=\frac{h}{x}\Rightarrow h=\sqrt{3x}..........(i)$$In $$\displaystyle \Delta ECD, \tan 30^{\circ}=\frac{ED}{CD}$$ $$\displaystyle \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{100-x}\Rightarrow h\sqrt{3}=100-x...........(ii)$$$$\therefore$$ Subst. the value of h from (i) in (ii) we get$$\displaystyle \sqrt{3x} \times \sqrt {3}=100-x\Rightarrow 3x=100-x\Rightarrow 4x=100\Rightarrow x=25m$$$$\therefore$$ $$\displaystyle h=(\sqrt{3}\times 25)=25\times 1.732m=43.3m$$$$\therefore$$ The required point is at a distance of $$25$$ m from the pillar B and the height of each pillar is $$43.3 m$$Mathematics

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