A boy standing in the middle of a field, observes a flying bird in the north at an angle of elevation of `30^(@)` and after 2 minutes, he observes the same bird in the south at an angle of elevation of `60^(@)`. If the bird flies all along in a straight line at a height of `50sqrt(3)` m, then its speed in km/h is

Options

A. 3
B. 9
C. 6
D. 4.5

Answer

To solve the problem step by step, we will analyze the situation using trigonometry and the information provided.

### Step 1: Understand the Problem
We have a boy standing in the middle of a field observing a bird at two different angles of elevation. The first observation is at an angle of elevation of \(30^\circ\) to the north, and the second observation is at an angle of elevation of \(60^\circ\) to the south. The bird is flying at a constant height of \(50\sqrt{3}\) meters.

### Step 2: Draw the Diagram
Draw a diagram with the boy at the center. Mark the north direction and the south direction. From the boy's position, draw two lines representing the angles of elevation to the bird:
- One line at \(30^\circ\) to the north.
- Another line at \(60^\circ\) to the south.

### Step 3: Calculate the Horizontal Distances
Using the height of the bird and the angles of elevation, we can find the horizontal distances to the bird from the boy's position.

1. **For the angle of elevation \(30^\circ\)**:
\[ \tan(30^\circ) = \frac{\text{Height}}{\text{Distance to the bird in the north}} \]
\[ \tan(30^\circ) = \frac{50\sqrt{3}}{d_1} \implies d_1 = \frac{50\sqrt{3}}{\tan(30^\circ)} = \frac{50\sqrt{3}}{\frac{1}{\sqrt{3}}} = 50 \cdot 3 = 150 \text{ meters} \]

2. **For the angle of elevation \(60^\circ\)**:
\[ \tan(60^\circ) = \frac{\text{Height}}{\text{Distance to the bird in the south}} \]
\[ \tan(60^\circ) = \frac{50\sqrt{3}}{d_2} \implies d_2 = \frac{50\sqrt{3}}{\tan(60^\circ)} = \frac{50\sqrt{3}}{\sqrt{3}} = 50 \text{ meters} \]

### Step 4: Calculate the Total Distance
The total distance the bird has flown from the north to the south is the sum of the two distances:
\[ \text{Total Distance} = d_1 + d_2 = 150 + 50 = 200 \text{ meters} \]

### Step 5: Calculate the Speed of the Bird
The bird takes 2 minutes to fly this distance. First, convert the time into seconds:
\[ \text{Time} = 2 \text{ minutes} = 2 \times 60 = 120 \text{ seconds} \]

Now, calculate the speed in meters per second:
\[ \text{Speed} = \frac{\text{Total Distance}}{\text{Time}} = \frac{200 \text{ meters}}{120 \text{ seconds}} = \frac{5}{3} \text{ m/s} \]

### Step 6: Convert Speed to km/h
To convert from meters per second to kilometers per hour, use the conversion factor \( \frac{18}{5} \):
\[ \text{Speed in km/h} = \frac{5}{3} \times \frac{18}{5} = 12 \text{ km/h} \]

### Final Answer
The speed of the bird is **12 km/h**.

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