Two poles of heights 10 m and 30 m stand vertically on level ground at a distance d apart. From a point on the ground between them, along the line joining their feet, the angles of elevation to their tops of the taller and the shorter poles are 60\(^\circ\) and 30\(^\circ\), respectively. The distance d between the poles is:
\(20\sqrt3\) meters
Let the point be at distance p from the taller pole (30 m, angle 60°) and \((d-p)\) from the shorter pole (10 m, angle 30°).
\(\tan60^\circ = \dfrac{30}{p} \Rightarrow p = \dfrac{30}{\sqrt3} = 10\sqrt3\).
\(\tan30^\circ = \dfrac{10}{d-p} \Rightarrow d-p = 10\sqrt3\).
Total distance: \(d = p+(d-p) = 10\sqrt3+10\sqrt3 = 20\sqrt3\) meters.
Hence, the distance d between the poles is \(20\sqrt3\) meters.
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