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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Question : If the sum and difference of two angles are 135° and $\frac{\pi}{12}$, respectively, then the value of the angles in degree measure is:
Option 1: 70° and 65°
Option 2: 75° and 60°
Option 3: 45° and 90°
Option 4: 80° and 55°
Question : An exterior angle of a triangle is 115° and one of the interior opposite angles is 45°. The other two angles are:
Option 1: 65° and 70°
Option 2: 60° and 75°
Option 4: 50° and 85°
Question : From the top of a cliff 100 metres high, the angles of depression of the top and bottom of a tower are 45° and 60°, respectively. The height of the tower is:
Option 1: $\frac{100}{3}(3-\sqrt{3})$ m
Option 2: $\frac{100}{3}(\sqrt3-1)$ m
Option 3: $\frac{100}{3}(2\sqrt3-1)$ m
Option 4: $\frac{100}{3}(\sqrt3-\sqrt{2})$ m
Question : Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from the two ships are 30° and 45°, respectively. If the lighthouse is 100 m high, the distance between the two ships is: (take $\sqrt{3}= 1.73$)
Option 1: 173 metres
Option 2: 200 metres
Option 3: 273 metres
Option 4: 300 metres
Question : The difference between the largest and the smallest angles of a triangle whose angles are in the ratio of 5 : 3 : 10 is:
Option 1: 20°
Option 2: 30°
Option 3: 50°
Option 4: 70°
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