To find the width of the river, a man observes the top of a tree, on the opposite bank making an angle of elevation of 60°. When he moves 24 meter backward from bank and observes the same top of the tree, his line of vision makes an angle of elevatio…

Answer: width of the river = 12 m, height of the tree = 20.76 m

Let the height of the tree be \(h\) and the width of the river be \(w\).

From the bank: \(\tan 60^\circ = \dfrac{h}{w}\), so \(h = w\sqrt{3}\) ... (I).

From 24 m further back: \(\tan 30^\circ = \dfrac{h}{w + 24}\), so \(\dfrac{1}{\sqrt{3}} = \dfrac{h}{w + 24}\), i.e. \(h = \dfrac{w + 24}{\sqrt{3}}\) ... (II).

From (I) and (II): \(w\sqrt{3} = \dfrac{w + 24}{\sqrt{3}}\), so \(3w = w + 24\) and \(w = 12\) m.

Then \(h = 12\sqrt{3} = 12 \times 1.73 = 20.76\) m. (Exactly, \(h \approx 20.78\) m.)