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A person standing on the bank of a river observes that the angle subtended by a tree on the opposite of bank is $60 °$ . When he reached $120 m$ behind the bank, he finds the angle to be $30 °$ . What is the breadth of the river?

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60 m

40 m

30 m

20 m

answer is A.

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Detailed Solution

It is given that a person who stands on the bank of a river observes the angle subtended by a tree on the opposite of bank is

60 °

.
According to the given data the figure has been drawn below.
Question Image

Let

R

be the breadth of river and

T

be the length of tree.
In

△ ABC

tan θ = P e r p e n d i c u l a r b a s e

\tan 60 ° = \frac{T}{R}

\Rightarrow T = \sqrt{3} R . . . . . . . (1)

(\tan 60 ° = \sqrt{3})

△ ABD

tan θ = P e r p e n d i c u l a r b a s e

\tan 30 ° = \frac{T}{R + 120}

\Rightarrow T = \frac{R + 120}{\sqrt{3}} . . . . . (2)

(\tan 30 ° = \frac{1}{\sqrt{3}})

From (1) and (2) equating the value of T we get,

\sqrt{3} R = \frac{R + 120}{\sqrt{3}}

\Rightarrow 3 R = R + 120

\Rightarrow 3 R - R = 120

\Rightarrow 2 R = 120

\Rightarrow R = \frac{120}{2}

\Rightarrow R = 60

Therefore, the breadth of the river is

60 m

.
So the correct option is 1.

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