A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is `60^(@)` . When he moves 50 m away from the bank, he finds the angle of elevation to be `30^(@)` . Calculate :
the width of the river and

Answer

To solve the problem step by step, we will use trigonometric relationships in right triangles formed by the man, the tree, and the points on the bank of the river.

### Step 1: Draw the Diagram
First, we need to visualize the situation. Draw a diagram where:
- Point A is the top of the tree.
- Point B is the position of the man on the bank of the river.
- Point C is directly below the tree on the opposite bank.
- Point D is the position of the man after he moves 50 meters away from the bank.

### Step 2: Label the Angles and Distances
- The angle of elevation from point B to point A (the tree) is \(60^\circ\).
- The angle of elevation from point D to point A is \(30^\circ\).
- The distance CD (the distance the man moved away from the bank) is 50 m.

### Step 3: Define the Variables
Let:
- \(h\) = height of the tree (AB)
- \(x\) = width of the river (BC)
- The distance from point D to point C (the distance from the man to the bank) is \(x + 50\).

### Step 4: Use Trigonometric Ratios
From triangle BCA (with angle \(60^\circ\)):
\[ \tan(60^\circ) = \frac{h}{x} \implies h = x \cdot \sqrt{3} \quad \text{(since } \tan(60^\circ) = \sqrt{3}\text{)} \]

From triangle DCA (with angle \(30^\circ\)):
\[ \tan(30^\circ) = \frac{h}{x + 50} \implies h = (x + 50) \cdot \frac{1}{\sqrt{3}} \quad \text{(since } \tan(30^\circ) = \frac{1}{\sqrt{3}}\text{)} \]

### Step 5: Set the Equations Equal
Since both expressions equal \(h\), we can set them equal to each other:
\[ x \cdot \sqrt{3} = (x + 50) \cdot \frac{1}{\sqrt{3}} \]

### Step 6: Clear the Fraction
Multiply both sides by \(\sqrt{3}\) to eliminate the fraction:
\[ 3x = x + 50 \]

### Step 7: Solve for \(x\)
Rearranging gives:
\[ 3x - x = 50 \implies 2x = 50 \implies x = 25 \]

### Conclusion
The width of the river (BC) is \(25 \, \text{m}\).