A boy standing on a horizontal plane finds a bird flying at a distane of 100 m from him at an elevation of `30^(@)`. A girl standing on the roof of a 20 m high building finds the angle of elevation of the same bird to be `45^(@)`. Boy and girl are on the opposite sides of the bird. Find the distance of the bird from the girl.

Answer

To solve the problem step by step, we will use trigonometric ratios to find the distance of the bird from the girl.

### Step 1: Understand the scenario
We have a boy (B) standing on a horizontal plane and a girl (G) on top of a 20 m high building. The bird (O) is flying at a distance of 100 m from the boy at an elevation angle of 30 degrees. The girl sees the bird at an elevation angle of 45 degrees.

### Step 2: Set up the diagram
1. Let O be the position of the bird.
2. Let B be the position of the boy.
3. Let G be the position of the girl on top of the building.
4. The height of the building (GF) is 20 m.

### Step 3: Find the height of the bird (OL)
Using the triangle OBL:
- OB = 100 m (distance from boy to bird)
- Angle of elevation (∠OBL) = 30 degrees

Using the sine function:
\[ \sin(30^\circ) = \frac{OL}{OB} \]
\[ \sin(30^\circ) = \frac{1}{2} \]
Substituting the values:
\[ \frac{1}{2} = \frac{OL}{100} \]
Cross-multiplying gives:
\[ OL = 100 \times \frac{1}{2} = 50 \text{ m} \]

### Step 4: Find the distance from the ground to the bird (OM)
Since the height of the building (GF) is 20 m, we can find OM (the vertical distance from the ground to the bird):
\[ OM = OL - GF = 50 \text{ m} - 20 \text{ m} = 30 \text{ m} \]

### Step 5: Find the distance from the bird to the girl (OG)
Using the triangle OMG:
- OM = 30 m (height from the ground to the bird)
- Angle of elevation (∠OGM) = 45 degrees

Using the sine function:
\[ \sin(45^\circ) = \frac{OM}{OG} \]
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \]
Substituting the values:
\[ \frac{1}{\sqrt{2}} = \frac{30}{OG} \]
Cross-multiplying gives:
\[ OG = 30 \sqrt{2} \text{ m} \]
Calculating the value:
\[ OG \approx 30 \times 1.414 = 42.42 \text{ m} \]

### Final Answer
The distance of the bird from the girl is approximately **42.42 m**.

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