Discussion
The pilot of a helicopter, at an altitude of 1200 m, finds that two ships are sailing towards it in the same direction. The angles of depression of the ships as observed from the helicopter are \( 60^{\circ} \) and \( 45^{\circ} \) respectively. Find the distance between the two ships.
Answer: C
Both ships lie on the same side of the helicopter, so the distance between them is the difference of their horizontal distances from the point directly below it.
\[ d_{60} = \frac{1200}{\tan 60^{\circ}} = \frac{1200}{\sqrt{3}} = 400\sqrt{3} = 692.8 \text{ m} \]
\[ d_{45} = \frac{1200}{\tan 45^{\circ}} = 1200 \text{ m} \]
\[ \text{separation} = 1200 - 400\sqrt{3} = 400\left(3 - \sqrt{3}\right) \]
\[ = 400 \times 1.268 = 507.2 \text{ m} \]
The steeper \( 60^{\circ} \) depression belongs to the nearer ship. Stopping at \( 400\sqrt{3} \) gives that ship's distance from the helicopter's base, not the gap between the two.
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