question_answer

The angles of depression of two boats as observed from the mast head of a ship 50 metres high are \[45{}^\circ \] and \[30{}^\circ \]. The distance between the boats, if they are on the same side of mast head in line with it, is

A) \[50\sqrt{3}\,\,m\]

B) \[50(\sqrt{3}+1)\,\,m\]

C) \[50(\sqrt{3}-1)\,\,m\]

D) \[5\left( 1-\frac{1}{\sqrt{3}} \right)\,\,m\]

Correct Answer: C

Solution :

 Let \[CD=x\] be the distance between the two boats. Then, from \[\Delta \,BCA,\]                                 \[\tan {{45}^{o}}=\frac{AB}{AC}\] or            \[1=\frac{50}{AD}\] or            \[AC=50\,m\] Again, from \[\Delta \,BDA,\]                 \[\tan {{30}^{o}}=\frac{AB}{AD}\] or            \[\frac{1}{\sqrt{3}}=\frac{50}{50+x}\] or            \[50+x=50\sqrt{3}\] or            \[x=50\,(\sqrt{3}-1)\,m\]