Two pillars of equal lengths stand on either side of a road which is $100$ m wide, exactly opposite to each other. At a point on the road between the pillars, the angles of elevation of the tops of th

Question

Two pillars of equal lengths stand on either side of a road which is $100$ m wide, exactly opposite to each other. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are $60°$ and $30°$. Find the length of each pillar and distance of the point on the road from the pillars. (Use $\sqrt{3} = 1.732$)

Accepted Answer

Step 1: Let $AB$ and $CD$ be two pillars of equal length $h$ m and let $P$ be the point on road $x$ m away from pillar $CD$.
Step 2: In $	riangle CDP$: $	an 60° = \sqrt{3} = \frac{h}{x} \Rightarrow h = \sqrt{3}x$ ... (i)
Step 3: In $	riangle ABP$: $	an 30° = \frac{1}{\sqrt{3}} = \frac{h}{100 - x} \Rightarrow h = \frac{100 - x}{\sqrt{3}}$ ... (ii)
Step 4: Solving (i) and (ii): $\sqrt{3}x = \frac{100 - x}{\sqrt{3}} \Rightarrow 3x = 100 - x \Rightarrow x = 25$
Step 5: $h = 25\sqrt{3} = 25 	imes 1.732 = 43.3$ m. Distance from pillars is $75$ m and $25$ m respectively.

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