From the top of a tower of 24 metre height, a man observes a child playing on the ground at an angle of depression of 30°. After some time, he observed that the angle of depression is 60°. Find the distance covered by the kid.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(16\sqrt{3}\) metre
The angle of depression from the top of the tower equals the angle of elevation of the top as seen from the child, because the two horizontal lines are parallel. In each right triangle with the tower, \(\tan\theta=\frac{\text{height}}{\text{distance}}\), so distance \(=\frac{\text{height}}{\tan\theta}\). The angle grows from 30° to 60°, so the child has moved towards the tower.
First position C1 (30°): \(\displaystyle BC_1=\frac{24}{\tan30^\circ}=24\sqrt{3}\) m
Second position C2 (60°): \(\displaystyle BC_2=\frac{24}{\tan60^\circ}=\frac{24}{\sqrt{3}}=8\sqrt{3}\) m
Distance covered: \(24\sqrt{3}-8\sqrt{3}=16\sqrt{3}\) m