An aeroplane at an altitude of 250 m observes the angle of depression of two boats on the opposite banks of a river to be `45^(@)and60^(@)` respectively. Find the width of the river. Write the answer correct to the nearest whole number.

Answer

To find the width of the river based on the angles of depression observed from an airplane, we can follow these steps:

### Step 1: Understand the Problem
An airplane is at an altitude of 250 m and observes two boats on opposite banks of a river at angles of depression of 45° and 60°. We need to find the width of the river.

### Step 2: Draw a Diagram
Draw a right triangle for each angle of depression. The altitude of the airplane forms the vertical side (250 m), and the horizontal distances to the boats form the base of the triangles.

### Step 3: Identify the Triangles
Let:
- Point A be the position of the airplane.
- Point B be the position of the boat observed at 45°.
- Point C be the position of the boat observed at 60°.

The angles of depression from point A to points B and C are 45° and 60°, respectively.

### Step 4: Use Trigonometric Ratios
For triangle AB (with angle 45°):
- The tangent of the angle is given by:
\[ \tan(45°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{250}{x} \]
Since \(\tan(45°) = 1\), we have:
\[ 1 = \frac{250}{x} \implies x = 250 \text{ m} \]

For triangle AC (with angle 60°):
- The tangent of the angle is given by:
\[ \tan(60°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{250}{y} \]
Since \(\tan(60°) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{250}{y} \implies y = \frac{250}{\sqrt{3}} \text{ m} \]

### Step 5: Calculate the Width of the River
The width of the river (W) is the sum of the distances x and y:
\[ W = x + y = 250 + \frac{250}{\sqrt{3}} \]
To simplify:
\[ W = 250 \left(1 + \frac{1}{\sqrt{3}}\right) \]

### Step 6: Calculate the Numerical Value
Using \(\sqrt{3} \approx 1.732\):
\[ W = 250 \left(1 + \frac{1}{1.732}\right) \approx 250 \left(1 + 0.577\right) \approx 250 \times 1.577 \approx 394.25 \text{ m} \]

### Step 7: Round to the Nearest Whole Number
The width of the river rounded to the nearest whole number is:
\[ W \approx 394 \text{ m} \]

### Final Answer
The width of the river is approximately **394 meters**.

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