From a point A on a bridge across a river, the angles of deression of the banks on opposite side of the river are `30^@ and 45^@`, respectively. If the bridge is at a height of 9 m from the suface of river, then find the width of the river.

Answer

To solve the problem of finding the width of the river given the angles of depression and the height of the bridge, we can follow these steps:

### Step 1: Understand the Geometry
- We have a bridge at height \( h = 9 \) m.
- The angles of depression to the banks of the river are \( 30^\circ \) and \( 45^\circ \).
- Let point A be the point on the bridge. The points on the opposite bank are B (for \( 30^\circ \)) and D (for \( 45^\circ \)).
- We need to find the width of the river, which is the distance \( CD \).

### Step 2: Draw the Diagram
- Draw a horizontal line representing the bridge at height 9 m.
- From point A, draw lines downwards at angles \( 30^\circ \) and \( 45^\circ \) to meet the water surface at points B and D respectively.

### Step 3: Use Trigonometry to Find Distances
1. **For angle \( 30^\circ \)**:
- In triangle \( ABC \):
- The height \( AB = 9 \) m (height of the bridge).
- Using the tangent function:
\[ \tan(30^\circ) = \frac{AB}{BC} \Rightarrow \frac{1}{\sqrt{3}} = \frac{9}{BC} \]
- Rearranging gives:
\[ BC = 9 \sqrt{3} \text{ m} \]

2. **For angle \( 45^\circ \)**:
- In triangle \( ABD \):
- Again, the height \( AB = 9 \) m.
- Using the tangent function:
\[ \tan(45^\circ) = \frac{AB}{BD} \Rightarrow 1 = \frac{9}{BD} \]
- Rearranging gives:
\[ BD = 9 \text{ m} \]

### Step 4: Calculate the Width of the River
- The total width \( CD \) of the river can be found by adding \( BC \) and \( BD \):
\[ CD = BC + BD = 9\sqrt{3} + 9 \]
- Approximating \( \sqrt{3} \approx 1.732 \):
\[ CD = 9(1.732) + 9 \approx 15.588 + 9 = 24.588 \text{ m} \]

### Final Answer
The width of the river is approximately \( 24.588 \) meters.

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