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 $60°$. If one ship is exactly behind the other on the same side of the lighth

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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 $60°$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. (Use $\sqrt{3} = 1.73$)

Accepted Answer

Step 1: Lighthouse PQ of height 75m with ships at S and R, angles of depression 30° and 60°
Step 2: $PQ = 75$ m. $\angle XQS = \angle QSP = 30°$, $\angle XQR = \angle QRP = 60°$. In $	riangle PQR$: $\frac{75}{PR} = 	an 60° = \sqrt{3} \Rightarrow PR = \frac{75}{\sqrt{3}} = 25\sqrt{3}$
Step 3: In $	riangle PQS$: $\frac{75}{PS} = 	an 30° \Rightarrow PS = 75\sqrt{3}$
Step 4: Distance $RS = PS - PR = 75\sqrt{3} - 25\sqrt{3} = 50\sqrt{3}$
Step 5: $= 50 	imes 1.73 = 86.5$ m

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