Two metal rods have coefficients of linear exapansion `1.1 xx 10^(-5 //)C` and `1.65 xx 10^(-5 //@)C` respectively. The difference in lengths is `10cm` at all temperatures. Their initial lengths must be respectively.

Options

A. `40 cm` and `50 cm`
B. `40 cm` and `30 cm`
C. `50 cm` and `60 cm`
D. `30 cm` and `20 cm`

Explanation

To solve the problem, we need to find the initial lengths of two metal rods given their coefficients of linear expansion and the fact that the difference in their lengths remains constant at 10 cm at all temperatures.

### Step-by-Step Solution:

1. **Define Variables**:
Let the initial lengths of the two rods be \( L_1 \) and \( L_2 \).
The coefficients of linear expansion are given as:
- \( \alpha_1 = 1.1 \times 10^{-5} \, \text{°C}^{-1} \)
- \( \alpha_2 = 1.65 \times 10^{-5} \, \text{°C}^{-1} \)

2. **Understand the Length Change**:
The change in length due to temperature change \( \Delta T \) can be expressed as:
- For rod 1: \( L_1' = L_1(1 + \alpha_1 \Delta T) \)
- For rod 2: \( L_2' = L_2(1 + \alpha_2 \Delta T) \)

3. **Set Up the Equation**:
According to the problem, the difference in lengths remains constant at 10 cm:
\[ L_1' - L_2' = 10 \, \text{cm} \]
Substituting the expressions for \( L_1' \) and \( L_2' \):
\[ L_1(1 + \alpha_1 \Delta T) - L_2(1 + \alpha_2 \Delta T) = 10 \]

4. **Rearranging the Equation**:
Expanding the equation:
\[ L_1 + L_1 \alpha_1 \Delta T - L_2 - L_2 \alpha_2 \Delta T = 10 \]
Rearranging gives:
\[ (L_1 - L_2) + (L_1 \alpha_1 - L_2 \alpha_2) \Delta T = 10 \]

5. **Using the Constant Difference**:
Since the difference \( L_1 - L_2 \) must also equal 10 cm (as stated in the problem), we can set:
\[ L_1 - L_2 = 10 \]
Thus, we can substitute this into our equation:
\[ 10 + (L_1 \alpha_1 - L_2 \alpha_2) \Delta T = 10 \]
This simplifies to:
\[ L_1 \alpha_1 - L_2 \alpha_2 = 0 \]

6. **Expressing One Length in Terms of the Other**:
From the equation \( L_1 \alpha_1 = L_2 \alpha_2 \):
\[ L_1 = \frac{\alpha_2}{\alpha_1} L_2 \]

7. **Substituting the Coefficients**:
Substitute the values of \( \alpha_1 \) and \( \alpha_2 \):
\[ L_1 = \frac{1.65 \times 10^{-5}}{1.1 \times 10^{-5}} L_2 \]
Simplifying gives:
\[ L_1 = 1.5 L_2 \]

8. **Using the Length Difference**:
Now, substitute \( L_1 \) back into the length difference equation:
\[ 1.5 L_2 - L_2 = 10 \]
This simplifies to:
\[ 0.5 L_2 = 10 \]
Therefore:
\[ L_2 = 20 \, \text{cm} \]

9. **Finding \( L_1 \)**:
Now, substitute \( L_2 \) back to find \( L_1 \):
\[ L_1 = 1.5 \times 20 = 30 \, \text{cm} \]

### Final Answer:
The initial lengths of the rods are:
- \( L_1 = 30 \, \text{cm} \)
- \( L_2 = 20 \, \text{cm} \)