The length of a rubber cord is `l_(1)` m when the tension is 4 N and `l_(2)` m when the tension is 6 N.The length when the tension is 9 N, is

Options

A. `(2.5l_(2) - 1.5l_(1))m`
B. `(6l_(2)- 1.5l_(1))m`
C. `(3l_(1) - l_(2))m`
D. `(3.5l_(2) =- 2.5l_(1))m`

Explanation

To solve the problem, we need to find the length of the rubber cord when the tension is 9 N, given the lengths at 4 N and 6 N. We can use the concept of linear elasticity and Young's modulus to relate the lengths and tensions.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two lengths of a rubber cord under different tensions. We denote:
- \( l_1 \) = length at 4 N
- \( l_2 \) = length at 6 N
- \( l_3 \) = length at 9 N (which we need to find)

2. **Using Young's Modulus**:
The formula for Young's modulus \( Y \) is given by:
\[ Y = \frac{F/A}{\Delta L / L_0} \]
where \( F \) is the force (tension), \( A \) is the cross-sectional area, \( \Delta L \) is the change in length, and \( L_0 \) is the original length.

3. **Setting Up the Equations**:
For the first tension (4 N):
\[ \frac{4}{A} = Y \cdot \frac{l_1 - L_0}{L_0} \]
For the second tension (6 N):
\[ \frac{6}{A} = Y \cdot \frac{l_2 - L_0}{L_0} \]

4. **Relating the Lengths**:
We can express the change in length for each case:
- \( \Delta L_1 = l_1 - L_0 \)
- \( \Delta L_2 = l_2 - L_0 \)
- \( \Delta L_3 = l_3 - L_0 \)

5. **Forming the Equations**:
From the equations of Young's modulus, we can derive:
\[ 4L_0 = A \cdot Y \cdot (l_1 - L_0) \quad (1) \]
\[ 6L_0 = A \cdot Y \cdot (l_2 - L_0) \quad (2) \]

6. **Solving for \( l_3 \)**:
We can set up a ratio using the tensions and lengths:
\[ \frac{l_1 - L_0}{4} = \frac{l_2 - L_0}{6} = \frac{l_3 - L_0}{9} \]

7. **Expressing \( l_3 \)**:
From the ratios, we can express \( l_3 \) in terms of \( l_1 \) and \( l_2 \):
\[ l_3 - L_0 = \frac{9}{4}(l_1 - L_0) \]
\[ l_3 - L_0 = \frac{9}{6}(l_2 - L_0) \]

8. **Finding the Lengths**:
Using the values of \( l_1 \) and \( l_2 \), we can calculate \( l_3 \). We can substitute the values of \( l_1 \) and \( l_2 \) into the equations derived.

9. **Final Calculation**:
After substituting and solving, we will find the value of \( l_3 \).

### Conclusion:
The length of the rubber cord when the tension is 9 N can be computed based on the derived relationships.