Q12 of 61 Page 215

From the top of a lighthouse of height 200 feet, the lighthouse keeper observes a Yacht and a Barge along the same line of sight . The angles of depression for the Yacht and the Barge are 45° and 30° respectively. For safety purposes the two sea vessels should be atleast 300 feet apart. If they are less than 300 feet, the keeper has to sound the alarm. Does the keeper have to sound the alarm?


Given, AB = 200 ft. and CD = ?


In triangle ABC,


∠ACB = ∠ OAC (alternate angles are equal)


= 45°


We know,





⇒ BC = 200 √2


And, in triangle ABD,


∠ADB = ∠OAD (alternate angles are equal)


= 30°




⇒ BD = 200 x 2


⇒ BD = 400


Now, CD = BD – BC


⇒ CD = 400 – 200 √2


= 200(2 – √2)


= 200 (2 – 1.414)


= 200(0.586)


= 117.2 m


∴distance between Yacht and a Barge = 117.2<300 m.


⇒ the keeper has to sound the alarm.


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