A man standing on the banks of a river observes that the angle subtended by a tree on the opposite bank is 60°. He walks 36 meters backward on the bank and observes the angle to be 30°. What is the breadth of the river?

Correct option is B

Given:

Initially, the angle of elevation to the top of the tree is 60°.
After walking 36 meters backward, the angle of elevation is 30°.
Formula Used:

Tan(ϴ) =perpendicular / base

Solution:

Let the breadth of the river =d meters,

let the height of the tree =h meters.

From the first observation:

​$$\tan(60^\circ) = \frac{h}{d}$$​​

​$$tan(60^\circ) = \sqrt{3}.$$​​

​$$\sqrt{3} = \frac{h}{d}$$​​

​$$h = d\sqrt{3}$$​​
From the second observation:
​$$\tan(30^\circ) = \frac{h}{d + 36}$$​​

​$$\tan(30^\circ) = \frac{1}{\sqrt{3}}$$​​
​$$\frac{1}{\sqrt{3}} = \frac{h}{d + 36}$$​​

Substitute h = d$$\sqrt{3}$$​into the second equation:

​$$\frac{1}{\sqrt{3}} = \frac{d\sqrt{3}}{d + 36}$$​​

Now, cross-multiply to solve for d:
d + 36 = 3d
2d = 36
d = 18
Thus, the breadth of the river is:18,meters.