To cross a river a person covers a straight forward distance of 325 m along a bridge over the river. If bridge substends `30^(@)` angle with the edge of the river, find the width of the river.

Answer

To solve the problem of finding the width of the river, we can use trigonometric ratios in a right triangle formed by the bridge and the river. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a bridge that a person crosses, which makes an angle of \(30^\circ\) with the edge of the river. The distance the person travels along the bridge is \(325\) meters. We need to find the width of the river.

### Step 2: Draw a Diagram
Draw a right triangle where:
- Point A is where the person starts on the bridge.
- Point B is directly across the river from point A.
- Point C is the point where the person ends up on the bridge.
- The angle \( \angle CAB = 30^\circ \).
- The length of AC (the distance along the bridge) is \(325\) m.
- The width of the river (AB) is what we need to find.

### Step 3: Identify the Right Triangle
In triangle ABC:
- AC is the hypotenuse (325 m).
- AB is the opposite side (the width of the river).
- Angle CAB is \(30^\circ\).

### Step 4: Use the Sine Function
We can use the sine function, which relates the opposite side to the hypotenuse in a right triangle:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, \( \theta = 30^\circ \), opposite = AB (width of the river), and hypotenuse = AC (325 m).

### Step 5: Set Up the Equation
Using the sine function:
\[ \sin(30^\circ) = \frac{AB}{325} \]
We know that \( \sin(30^\circ) = \frac{1}{2} \).

### Step 6: Substitute and Solve for AB
Substituting the value of sine:
\[ \frac{1}{2} = \frac{AB}{325} \]
Now, multiply both sides by \(325\):
\[ AB = 325 \times \frac{1}{2} = 162.5 \text{ m} \]

### Step 7: Conclusion
The width of the river is \(162.5\) meters.

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