A boy standing on a horizontal plane find that angle of elevation of a bird 100 meter away from him at `30^(@)`. A girl standing at a house 20 meter above the plane find that elevation of the bird is `45^(@)`. If boy and girl are in the opposite direction find the distance between the bird and the girl.

Answer

To solve the problem step by step, we will use trigonometric principles related to angles of elevation and the properties of right triangles.

### Step 1: Understand the problem
We have a boy standing on a horizontal plane who sees a bird at an angle of elevation of \(30^\circ\) from a distance of 100 meters. A girl is standing on a house that is 20 meters above the horizontal plane, and she sees the same bird at an angle of elevation of \(45^\circ\). We need to find the distance between the bird and the girl.

### Step 2: Calculate the height of the bird from the boy's perspective
Using the angle of elevation of \(30^\circ\) and the distance of 100 meters from the boy to the point directly below the bird, we can use the tangent function:

\[ \tan(30^\circ) = \frac{\text{height of the bird (h)}}{\text{distance from the boy to the point below the bird (100 m)}} \]

From trigonometric values, we know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

So we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{h}{100} \]

Solving for \(h\):
\[ h = 100 \cdot \frac{1}{\sqrt{3}} \approx 57.74 \text{ meters} \]

### Step 3: Calculate the height of the bird from the girl's perspective
Now, we consider the girl who is 20 meters above the horizontal plane. The angle of elevation to the bird is \(45^\circ\). We can again use the tangent function:

\[ \tan(45^\circ) = \frac{\text{height of the bird above the girl (h - 20)}}{\text{horizontal distance from the girl to the point below the bird (d)}} \]

From trigonometric values, we know:
\[ \tan(45^\circ) = 1 \]

This gives us:
\[ 1 = \frac{h - 20}{d} \]

So, we can express \(d\) in terms of \(h\):
\[ d = h - 20 \]

### Step 4: Substitute the height of the bird
Now, substituting the value of \(h\) we found earlier:
\[ d = 57.74 - 20 = 37.74 \text{ meters} \]

### Step 5: Calculate the total distance between the girl and the bird
Since the boy and girl are in opposite directions, the total distance between the girl and the bird is the sum of the distance from the boy to the bird and the distance from the girl to the bird:

\[ \text{Total distance} = 100 + d = 100 + 37.74 = 137.74 \text{ meters} \]

### Final Answer
The distance between the bird and the girl is approximately **137.74 meters**.

---