A mass of 10kg is suspended vertically by a rope of length 5 m from the roof. A force of 30 N is applied at the middle point of rope in horizontal direction .The angle made by upper half of the rope with vertical is `theta = tan^(-1) ( x xx 10^(-1))`. The value of x is _______
(Given , g ` = 10 m//s^(2)`)

Explanation

To solve the problem, we need to analyze the forces acting on the mass suspended by the rope and the effect of the horizontal force applied at the midpoint of the rope.

### Step-by-Step Solution:

1. **Identify the Forces**:
- The mass \( m = 10 \, \text{kg} \) exerts a downward force due to gravity, which is \( mg = 10 \times 10 = 100 \, \text{N} \).
- A horizontal force \( F = 30 \, \text{N} \) is applied at the midpoint of the rope.

2. **Set Up the Geometry**:
- The rope has a total length of \( L = 5 \, \text{m} \). The midpoint is at \( 2.5 \, \text{m} \).
- The angle \( \theta \) is formed by the upper half of the rope with the vertical.

3. **Equilibrium Conditions**:
- For the horizontal direction, the tension component must balance the applied force:
\[ T \sin \theta = F \]
- For the vertical direction, the tension component must balance the weight of the mass:
\[ T \cos \theta = mg \]

4. **Substituting Known Values**:
- From the vertical equilibrium equation:
\[ T \cos \theta = 100 \quad \text{(1)} \]
- From the horizontal equilibrium equation:
\[ T \sin \theta = 30 \quad \text{(2)} \]

5. **Dividing the Equations**:
- Divide equation (2) by equation (1):
\[ \frac{T \sin \theta}{T \cos \theta} = \frac{30}{100} \]
- This simplifies to:
\[ \tan \theta = \frac{30}{100} = \frac{3}{10} \]

6. **Finding \( \theta \)**:
- We know that \( \theta = \tan^{-1} \left( \frac{3}{10} \right) \).

7. **Relating to the Given Equation**:
- The problem states that \( \theta = \tan^{-1} \left( \frac{x}{10} \right) \).
- Therefore, we can equate:
\[ \frac{x}{10} = \frac{3}{10} \]

8. **Solving for \( x \)**:
- Multiply both sides by 10:
\[ x = 3 \]

### Final Answer:
The value of \( x \) is \( \boxed{3} \).