The angle of elevation of the top of a tower from two points on opposite sides of the tower are 30° and 60°. If the distance between the two points is 54 m, find the height of the tower.
\(13.5\sqrt{3}\) m
Let the height be h and the distances of the two points from the foot be x and y.
tan 30° = h/x gives x = h\(\sqrt{3}\); tan 60° = h/y gives y = \(\frac{h}{\sqrt{3}}\).
The points are on opposite sides, so x + y = 54: h\(\sqrt{3}\) + \(\frac{h}{\sqrt{3}}\) = \(\frac{4h}{\sqrt{3}}\) = 54.
h = \(\frac{54\sqrt{3}}{4}\) = \(13.5\sqrt{3}\) m.
Hence, the answer is \(13.5\sqrt{3}\) m.
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