question_answer

A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is \[60{}^\circ .\] when he retreats 20 feet from the bank, he finds the angle to be\[30{}^\circ \]. The breadth of the river in feet is:

A) 15

B) \[15\sqrt{3}\]

C) \[10\sqrt{3}\]

D) 10

Correct Answer: D

Solution :

[d] Let h be the height of tree PQ and breadth of river PS be x ft. Angle of elevation subtended by a tree is \[60{}^\circ \]. Also, when he retreats 20 feet, the angle becomes\[30{}^\circ \]. Also, in \[\Delta PQS,\tan 60{}^\circ =\frac{h}{x}\Rightarrow h=\sqrt{3}x\] and in \[\Delta PQR,\tan 30{}^\circ =\frac{h}{x+20}\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x+20}\] \[\Rightarrow x+20=\sqrt{3}h\] \[\Rightarrow x+20=3x\] (By putting value of h) \[\Rightarrow 2x=20\Rightarrow x=10\] Hence breadth of river is 10 ft.