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 returns 40 m from the bank, he finds the angle to be `30^(@).` What is the breadth of the river?

Options

A. A) 60 m
B. B) 40 m
C. C) 30 m
D. D) 20 m

Answer

To solve the problem step by step, we will use trigonometric principles related to right triangles.

### Step 1: Understanding the Setup
Let’s denote:
- \( x \) = breadth of the river (the distance from the person to the base of the tree).
- \( h \) = height of the tree.

The person first stands at the bank of the river and sees the top of the tree at an angle of \( 60^\circ \). When he moves back 40 meters, the angle becomes \( 30^\circ \).

### Step 2: Setting Up the First Equation
From the first position (at the bank):
- We can use the tangent function:
\[ \tan(60^\circ) = \frac{h}{x} \]
We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h}{x} \implies h = \sqrt{3}x \quad \text{(Equation 1)} \]

### Step 3: Setting Up the Second Equation
From the second position (40 meters back):
- The distance from the person to the tree is now \( x + 40 \).
- Again using the tangent function:
\[ \tan(30^\circ) = \frac{h}{x + 40} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x + 40} \implies h = \frac{1}{\sqrt{3}}(x + 40) \quad \text{(Equation 2)} \]

### Step 4: Equating the Two Expressions for \( h \)
Now, we have two expressions for \( h \):
1. From Equation 1: \( h = \sqrt{3}x \)
2. From Equation 2: \( h = \frac{1}{\sqrt{3}}(x + 40) \)

Setting them equal:
\[ \sqrt{3}x = \frac{1}{\sqrt{3}}(x + 40) \]

### Step 5: Cross-Multiplying to Solve for \( x \)
Cross-multiplying gives:
\[ \sqrt{3} \cdot \sqrt{3} x = x + 40 \]
This simplifies to:
\[ 3x = x + 40 \]

### Step 6: Isolating \( x \)
Now, we can isolate \( x \):
\[ 3x - x = 40 \implies 2x = 40 \implies x = 20 \]

### Conclusion
Thus, the breadth of the river is:
\[ \boxed{20 \text{ meters}} \]