Question : A man standing on the bank of a river observes that the angle of elevation of the top of a tree just on the opposite bank is $60°$. But the angle of elevation is $30°$ from a point which is at a distance $20\sqrt{3}$ ft away from the bank. Then the height of the tree is:
Option 1: 60 ft
Option 2: 45 ft
Option 3: 30 ft
Option 4: 15 ft
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Correct Answer: 30 ft
Solution : Let $h$ be the height of the tree and $x$ be the breadth of the river. From the above figure we get, $\tan60°=\frac{h}{x}$ ⇒ $\sqrt{3}=\frac{h}{x}$ ⇒ $x=\frac{h}{\sqrt{3}}$----------(equation 1) Also, $\tan30°=\frac{h}{x+20\sqrt{3}}$ ⇒ $\frac{1}{\sqrt{3}}=\frac{h}{x+20\sqrt{3}}$ ⇒ $x=h\sqrt{3}–20\sqrt{3}$ ---------(equation 2) Now equalising the values of $x$ from both equations we get, ⇒ $h\sqrt{3}–20\sqrt{3}=\frac{h}{\sqrt{3}}$ ⇒ $3h–60=h$ ⇒ $2h=60$ ⇒ $h=30$ ft Hence, the correct answer is 30 ft.
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