As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

A. 20√3m. B. 18√3m. C. 16√3m. D. 22√3m. Answer: Option A
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Solution(By Apex Team)

Height and Distance solution image In Right triangle, $\Delta C D O$ $\begin{array}{l}\tan30^{\circ}=\frac{CD}{OD}\\ \Rightarrow\frac{1}{\sqrt{3}}=\frac{CD}{OD}\\ \Rightarrow CD=\frac{OD}{\sqrt{3}}\ldots(i)\end{array}$ Also, In Right Angle; $\triangle A B O,$ $\begin{array}{l} \tan 60^{\circ}=\frac{A B}{O B} \\ \Rightarrow \sqrt{3}=\frac{A B}{(80-O D)} \\ \Rightarrow A B=\sqrt{3}(80-O D) \\ A B=C D \\ \Rightarrow \sqrt{3}(80-O D)=\frac{O D}{\sqrt{3}} \\ \Rightarrow 3(80-O D)=O D \\ \Rightarrow 240-3 O D=O D \\ \Rightarrow 4 O D=240 \\ \Rightarrow O D=60 \mathrm{~m} \end{array}$ Putting the value of OD in equation (i) $\begin{array}{l}CD=\frac{OD}{\sqrt{3}}\\ \Rightarrow CD=\frac{60}{\sqrt{3}}\\ \Rightarrow CD=20\sqrt{3}m\end{array}$