Discussion
A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60°. When he retreats 40 m from the bank, he finds the angle to be 30°. The height of the tree is:
Answer: B
Let h be the height of the tree and x be the initial distance from the tree.
From the first observation: \(\tan 60° = \frac{h}{x} \Rightarrow \sqrt{3} = \frac{h}{x} \Rightarrow h = x\sqrt{3}\).
From the second observation: \(\tan 30° = \frac{h}{x+40} \Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{x+40} \Rightarrow x+40 = h\sqrt{3}\).
Substitute h from the first equation into the second: \(x+40 = (x\sqrt{3})\sqrt{3} = 3x\).
\(40 = 2x \Rightarrow x = 20\) m.
Now find the height: \(h = x\sqrt{3} = 20\sqrt{3}\) m.
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