Question

A man standing on the bank of river observes that the angle subtended by a tree on the opposite bank is 60${}^0$. When he retires 36 m from the bank, he finds the angle to be 30${}^0$. Find the breadth of the river.

Answer 1

The above diagram can be drawn from the given information.

Let AB be the tree and AC be the river.

Also, let C and D be the two positions of the man.

Then, $\angle ACB={60}^0,\angle ADB={30}^0$ and $CD=36m$ [Given]
Then $AD=(36+x)m$
In $△ABD,$

$\frac{AB}{AD}=\tan {30}^0$

$\Rightarrow \frac{h}{36+x}=\frac{1}{\sqrt{3}}[\because \tan {30}^o=\frac{1}{\sqrt{3}}]$

$\Rightarrow h=\frac{36+x}{\sqrt{3}}$ ....(i)

Also, in $△ABC,$

$\frac{AB}{AC}=\tan {60}^0$

$\Rightarrow \frac{h}{x}=\sqrt{3}[\because \tan {60}^o=\sqrt{3}]$

$\Rightarrow h=\sqrt{3x}$ ....(ii)

From (i) and (ii), we get:

$\frac{36+x}{\sqrt{3}}=\sqrt{3x}$

$\Rightarrow x=18m$

Hence, the breadth of the river is $18m$.