Question : An observer on the top of a mountain, 500 m above sea level, observes the angles of depression of the two boats in his same place of vision to be 45° and 30°, respectively. Then the distance between the boats, if the boats are on the same side of the mountain, is:
Option 1: 456 m
Option 2: 584 m
Option 3: 366 m
Option 4: 699 m
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Correct Answer: 366 m
Solution :
Given:
AB = Height of mountain = 500 m
$\angle$ACB = 30°; $\angle$ADB = 45°
C and D ⇒ Positions of boats
Let CD = $x$ m
Solution:
From $\triangle$ABD,
$\tan 45° = \frac{AB}{BD}$
⇒ AB = BD
⇒ AB = BD = 500 m
In $\triangle$ABC, we have
$\tan 30° = \frac{AB}{BC}$
⇒ $\frac{1}{\sqrt{3}}$ = $\frac{500}{500+x}$
⇒ 500 + $x$ = 500$\sqrt{3}$
⇒ $x$ = 500$\sqrt{3}$–500
⇒ $x$ = 500 ($\sqrt{3}$–1) m
⇒ $x$ = 500 (1.732–1) m
⇒ $x$ = (500 × 0.732) m
⇒ $x$ = 366 metres
Hence, the correct answer is 366 m.
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