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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