From a point an a bridge across river, the angles of depression of the banks on opposite sides of the river are `30^(@) and 45^(@)` respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.

Answer

To find the width of the river based on the given angles of depression and the height of the bridge, we can follow these steps:

### Step 1: Understand the Setup
We have a bridge at a height of 3 meters above the banks of the river. From a point on the bridge, the angles of depression to the banks on either side of the river are given as 30 degrees and 45 degrees.

### Step 2: Identify the Triangles
We can visualize two right triangles:
1. Triangle ACB for the bank at 45 degrees.
2. Triangle AED for the bank at 30 degrees.

### Step 3: Calculate the Distance to the Bank at 45 Degrees
Using triangle ACB:
- The angle of depression is 45 degrees.
- The height of the bridge (perpendicular) is 3 meters.

Using the tangent function:
\[ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} \]
Since \(\tan(45^\circ) = 1\):
\[ 1 = \frac{3}{AC} \implies AC = 3 \text{ meters} \]

### Step 4: Calculate the Distance to the Bank at 30 Degrees
Using triangle AED:
- The angle of depression is 30 degrees.
- The height of the bridge (perpendicular) is still 3 meters.

Using the tangent function:
\[ \tan(30^\circ) = \frac{DE}{DA} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{3}{DA} \implies DA = 3\sqrt{3} \text{ meters} \]

### Step 5: Calculate the Width of the River
The total width of the river (W) is the sum of the distances AC and DA:
\[ W = AC + DA = 3 + 3\sqrt{3} \]

### Final Answer
Thus, the width of the river is:
\[ W = 3 + 3\sqrt{3} \text{ meters} \]