Two poles of equal height stand on either side of a road 80 m wide, exactly opposite each other. From a point on the road between them, the angles of elevation of their tops are 60° and 30°. Find the height of the poles and the distance of the point…

Answer: Height \(= 20\sqrt3 = 34.6\) m; the point is 20 m from the first pole and 60 m from the second.

[Figure: Poles AB and CD of height h, feet B and D, BD = 80 m. Point P on BD with BP = x, PD = 80 – x. \(\angle APB = 60^\circ\), \(\angle CPD = 30^\circ\).]

In ΔABP: \(\tan 60^\circ = \dfrac{h}{x}\) \(\therefore h = \sqrt3\,x\) ...(I)

In ΔCDP: \(\tan 30^\circ = \dfrac{h}{80 - x}\) \(\therefore h = \dfrac{80 - x}{\sqrt3}\) ...(II)

From (I) and (II): \(\sqrt3\,x = \dfrac{80 - x}{\sqrt3}\) \(\therefore 3x = 80 - x\) \(\therefore 4x = 80\) \(\therefore x = 20\)

\(h = 20\sqrt3 = 20 \times 1.73 = 34.6\) m. The point is 20 m from pole AB and 80 – 20 = 60 m from pole CD.