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A man standing on the top of a 150 m high tower observes the angles of depression of two cars on opposite sides of the tower to be $30^\circ$ and $60^\circ$. What is the distance between the two cars?

1. 200 $\sqrt{3}$ m
2. 150 $\sqrt{3}$ m
3. 100 $\sqrt{3}$ m
4. 300 m

Correct answer: 200 $\sqrt{3}$ m

Solution

For the car seen at $30^\circ$, horizontal distance from the tower is $150\cot 30^\circ = 150\sqrt{3}$. For the car seen at $60^\circ$, it is $150\cot 60^\circ = 50\sqrt{3}$. Since they are on opposite sides, the distance between them is $150\sqrt{3}+50\sqrt{3}=200\sqrt{3}$ m.

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