# A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $\dpi{100} 60^{\circ}$ and when he retires 40 meters away from the tree the angle of   elevation becomes $\dpi{100} 30^{\circ}$. The breadth of the river is   Option 1) $40m$ Option 2) $30 m$ Option 3) $20m$ Option 4) $60m$

S Sabhrant Ambastha

As we learnt in

Height and Distances -

The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

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Let OB represent bank, OA=height=h

$In \bigtriangleup OAB, \: \tan 60^{\circ}=\frac{OA}{OB}\Rightarrow OB=\frac{h}{\sqrt{3}}$

$Similarly, in \bigtriangleup OAC, \tan 30^{\circ}= \frac{OA}{OC}\Rightarrow OC=\sqrt{3}h$

$Now, BC=OC-OB=h\left [ \sqrt{3}-\frac{1}{\sqrt{3}} \right ]=\frac{2}{\sqrt{3}}h$

$h=\frac{\sqrt{3}}{2}\times 40= 20\sqrt{3}$

$Now, OB=\frac{h}{\sqrt{3}}=\frac{20\sqrt{3}}{\sqrt{3}}= 20m$

Option 1)

$40m$

This option is incorrect

Option 2)

$30 m$

This option is incorrect

Option 3)

$20m$

This option is correct

Option 4)

$60m$

This option is incorrect

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