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A tall tree stands vertically on a bank of a river. At the point on the other bank directly opposite the tree, the angle of elevation of the top of the tree is $60 °$ . At a point 20 m behind this point on the same bank, the angle of elevation of the top of the tree is $30 °$ what will be the height of the tree and the width of the river?

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103 m

203 m

153 m

102 m

answer is A.

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Detailed Solution

Given,
The angle of elevation of the top of the tree =

60 °

Distance of the point= 20 m
The angle of elevation of the top of the tree =

30 °

We use the trigonometric identity

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

.
We draw the diagram,
Question Image

We get the value of BC from

Δ A C B

tan 30 ° = A B B C ⇒ 13 = A B x [∵ tan 30 ° = 13] ⇒ x = 3 A B ....... (i)

We get the value of DB from

Δ A B D

tan 60 ° = A B D B ⇒ 3 = A B x − 20 [∵ tan 60 ° = 3] ⇒ A B = 3 x − 20

⇒ A B = 3 x − 203 ⇒ A B + 203 = 3 x ⇒ x = A B + 203 3 ........ (i i)

We determine the value of h by using the equations (i) and (ii),

⇒ 3 A B = A B + 203 3 ⇒ A B 3 = A B + 203 3 ⇒ 3 A B = A B + 203 ⇒ 2 A B = 203

⇒ A B = 103

m
Thus the height of tree will be

103 m

.
Therefore the correct option is 1.

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