Roshani saw an eagle on the top of a tree at an angle of elevation of `61^(@)`, while she was standing at the door of her house. She went on the terrace of the house so that she could see it clearly. The terrace was at a height of `4m`. While observing the eagle from there the angle of elevation was `52^(@)`. At what height from the ground was the eagle? `tan 61^(@)=1.8, tan 52^(@)=1.28, tann 29^(@)=0.55, tan 38^(@)=0.78)`

Answer

To find the height of the eagle from the ground, we can break down the problem into steps using trigonometric relationships.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Roshani first sees the eagle at an angle of elevation of \(61^\circ\) from the ground level.
- After moving to the terrace, which is \(4m\) high, she sees the eagle at an angle of elevation of \(52^\circ\).
- We need to find the total height of the eagle from the ground.

2. **Setting Up the Diagram**:
- Let \(RS\) be the height of the eagle from the ground.
- Let \(Y\) be the height of the tree above the terrace (i.e., the height of the eagle above the terrace).
- Therefore, the total height of the eagle from the ground can be expressed as \(RS = Y + 4\).

3. **Using the First Angle of Elevation**:
- From the ground level, using the angle of elevation \(61^\circ\):
\[ \tan(61^\circ) = \frac{Y + 4}{X} \]
where \(X\) is the horizontal distance from Roshani to the base of the tree.
- Given \( \tan(61^\circ) = 1.8 \), we can write:
\[ 1.8 = \frac{Y + 4}{X} \quad \text{(1)} \]

4. **Using the Second Angle of Elevation**:
- From the terrace, using the angle of elevation \(52^\circ\):
\[ \tan(52^\circ) = \frac{Y}{X} \]
- Given \( \tan(52^\circ) = 1.28 \), we can write:
\[ 1.28 = \frac{Y}{X} \quad \text{(2)} \]

5. **Expressing \(X\) in Terms of \(Y\)**:
- From equation (2), we can express \(X\) as:
\[ X = \frac{Y}{1.28} \quad \text{(3)} \]

6. **Substituting \(X\) in Equation (1)**:
- Substitute equation (3) into equation (1):
\[ 1.8 = \frac{Y + 4}{\frac{Y}{1.28}} \]
- Simplifying gives:
\[ 1.8 \cdot \frac{Y}{1.28} = Y + 4 \]
- Rearranging leads to:
\[ 1.8Y = 1.28(Y + 4) \]
- Expanding and simplifying:
\[ 1.8Y = 1.28Y + 5.12 \]
\[ 1.8Y - 1.28Y = 5.12 \]
\[ 0.52Y = 5.12 \]
\[ Y = \frac{5.12}{0.52} = 9.85 \text{ meters} \]

7. **Calculating the Total Height of the Eagle**:
- Now substitute \(Y\) back into the equation for \(RS\):
\[ RS = Y + 4 = 9.85 + 4 = 13.85 \text{ meters} \]

### Final Answer:
The height of the eagle from the ground is **13.85 meters**.