Two pillars of equal lengths stand on either side of a road which is $\displaystyle 100$ 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 $\displaystyle 60$° and $\displaystyle 30$°. Find the length of each pillar and distance of the point on the road from the pillars. (Use $\displaystyle \sqrt{3}=1 \cdot 732$ )

Let AB and CD are two pillars of equal length h m and let P be the point on road x m away from pillar CD. In \(\displaystyle \triangle \mathrm{CDP}\) \[\begin{aligned} & \tan 60^{\circ}=\sqrt{3}=\frac{\mathrm{h}}{\mathrm{x}} \\ & \Rightarrow \mathrm{~h}=\sqrt{3} \times \end{aligned} \] In \(\displaystyle \triangle \mathrm{ABP}\), \[\tan 30^{\circ}=\frac{1}{\sqrt{3}}=\frac{\mathrm{h}}{100-\mathrm{x}} \] \[\Rightarrow \mathrm{h}=\frac{100-x}{\sqrt{3}} \] Solving eq.(i) and eq.(ii) \[x=25 \] and \(\displaystyle \mathrm{h}=25 \sqrt{3}=25 \times 1.732=43.3\) The length of each pillar is $\displaystyle 43.3$ m and the distance of the point on the road from pillars is $\displaystyle 75$ m and $\displaystyle 25$ m respectively.