From the top of a cliff 80 meters high, the angles of depression of two ships in the same direction are 30o and 60o. Find the distance between the ships (rounded off to two decimal places).
92.38 meters
Let the cliff be \(AB = 80\) m, with the two ships at points C and D on the ground, in the same direction from the base B.
For the ship with angle of depression \(60^\circ\) (nearer ship, at D): \(\tan 60^\circ = \frac{AB}{BD}\), so \(BD = \frac{80}{\sqrt{3}} = 46.19\) m.
For the ship with angle of depression \(30^\circ\) (farther ship, at C): \(\tan 30^\circ = \frac{AB}{BC}\), so \(BC = 80\sqrt{3} = 138.56\) m.
Distance between the ships: \(CD = BC - BD = 138.56 - 46.19 = 92.38\) m.
Hence, the distance between the ships is 92.38 meters.
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