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Heights and Distances - 3D Trigonometry

XAT 2018 · Quantitative Ability & Data Interpretation · Medium · Quantitative Aptitude

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A boat, stationed at the North of a lighthouse, is making an angle of 30° with the top of the lighthouse. Simultaneously, another boat, stationed at the East of the same lighthouse, is making an angle of 45° with the top of the lighthouse. What will be the shortest distance between these two boats? The height of the lighthouse is 300 feet. Assume both the boats are of negligible dimensions.

  1. A.

    300 feet

  2. B.

    6003\frac{600}{\sqrt{3}} feet

  3. C.

    3003\frac{300}{\sqrt{3}} feet

  4. D.

    600 feet

  5. E.

    None of the above

Answer

D

Explanation

Distance to East boat d1=300tan⁡45∘=300d_1 = \frac{300}{\tan 45^\circ} = 300 feet. Distance to North boat d2=300tan⁡30∘=3003d_2 = \frac{300}{\tan 30^\circ} = 300\sqrt{3} feet. Since North and East are perpendicular, distance between boats = d12+d22=3002+(3003)2=3002(1+3)=300×2=600\sqrt{d_1^2 + d_2^2} = \sqrt{300^2 + (300\sqrt{3})^2} = \sqrt{300^2(1 + 3)} = 300 \times 2 = 600 feet.

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