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A tree is broken by the wind and now its upper part touches the ground at a point $10$ m from the foot of the tree and makes an angle of $60 °$ with the ground. The entire length of the tree is

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15

20

(10 + \frac{20}{\sqrt{3}}) \sqrt{3} m

(10 + \frac{\sqrt{3}}{2}) m

answer is C.

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Detailed Solution

It is given that the tree is broken by the wind and its upper part touches the ground at a point

10

m from the foot of the tree which makes an angle of 60

°

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

Let, say the Length of the tree

= BT = BC + CT

and

CT = CA

when the tree is broken.
Given A

B = 10 m

.
In

△ ABC

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

\tan 60 ° = \frac{BC}{AB}

............(1)
Also,

sec θ = H y p o t e n u s e b a s e

\sec 60 ° = \frac{AC}{AB}

...................(2)

BT =

BC + AC

\Rightarrow

ABtan 60 ° + AB sec 60 °

[Using (1) and (2)]

\Rightarrow AB (\sqrt{3)} + AB (2)

∵ tan 60 ° = 3, sec 60 ° = 2

\Rightarrow AB (\sqrt{3} + 2)

\Rightarrow 10 (1 + \frac{2}{\sqrt{3}}) \sqrt{3} m

\Rightarrow (10 + \frac{20}{\sqrt{3}}) \sqrt{3}

m
So the entire length of the tree is

(10 + \frac{20}{\sqrt{3}}) \sqrt{3}

m.
So, the correct option is 3.

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