Question : The cliff of a mountain is 180 m high and the angles of depression of two ships on either side of the cliff are 30° and 60°. What is the distance between the two ships?
Option 1: $400$ metres
Option 2: $400\sqrt{3}$ metres
Option 3: $415.68$ metres
Option 4: $398.6$ metres
Correct Answer: $415.68$ metres
Solution : AD = Cliff = 180 metres $\angle$ABD = 60°, $\angle$ACD = 30° From ∆ABD, $\tan 60° = \frac{AD}{BD}$ ⇒$\sqrt{3}=\frac{180}{BD}$ $\therefore BD=60\sqrt{3}$ metres From ∆ACD, $\tan 30° = \frac{AD}{CD}$ ⇒ $\frac{1}{\sqrt{3}}=\frac{180}{CD}$ $\therefore CD=180\sqrt{3}$ metres ∴ BC = BD + DC $\therefore BC=60\sqrt{3}+180\sqrt{3} =240\sqrt{3}=(240 × 1.732)= 415.68$ metres So, the distance between the two ships is $415.68$ metres. Hence, the correct answer is $415.68$ metres.
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