A watchman at the top of a 30 m high tower looks at two stones lying on the ground, one on each side of the tower, in a straight line with its foot. The angles of depression of the stones are \(45^\circ\) and \(60^\circ\). How far apart are the two stones?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(10(3+\sqrt{3})\) m
An angle of depression from the top equals the angle the line of sight makes with the ground at the stone. Each stone gives a right-angled triangle with the 30 m tower opposite that angle, so \(\text{distance}=\displaystyle\frac{30}{\tan(\text{angle})}\). The stones are on opposite sides, so the two distances are added.
Stone seen at \(45^\circ\): distance \(=\displaystyle\frac{30}{\tan 45^\circ}=30\) m
Stone seen at \(60^\circ\): distance \(=\displaystyle\frac{30}{\sqrt{3}}=10\sqrt{3}\) m
Distance between the stones \(=30+10\sqrt{3}=10(3+\sqrt{3})\) m
