From the top of a 60 m high tower, a guard looks at two cars parked on the level ground, one on each side of the tower, in a straight line with its foot. The angles of depression of the cars are \(30^\circ\) and \(60^\circ\). How far apart are the two cars?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(80\sqrt{3}\) m
An angle of depression from the top equals the angle the line of sight makes with the ground at the car. Each car gives a right-angled triangle with the 60 m tower opposite that angle, so \(\text{distance}=\displaystyle\frac{60}{\tan(\text{angle})}\). The cars are on opposite sides, so the two distances are added.
Car seen at \(30^\circ\): distance \(=\displaystyle\frac{60}{1/\sqrt{3}}=60\sqrt{3}\) m
Car seen at \(60^\circ\): distance \(=\displaystyle\frac{60}{\sqrt{3}}=20\sqrt{3}\) m
Distance between the cars \(=60\sqrt{3}+20\sqrt{3}=80\sqrt{3}\) m
