To solve the problem, we need to find the length of the aluminum rod given that the increase in length of both the copper and aluminum rods is independent of the increase in temperature. We will use the formula for linear expansion of materials.
### Step-by-Step Solution:
1. **Understand the Linear Expansion Formula**:
The formula for linear expansion is given by:
\[
L = L_0 (1 + \alpha \Delta T)
\]
where:
- \( L \) is the final length,
- \( L_0 \) is the original length,
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta T \) is the change in temperature.
2. **Set Up the Equations for Both Rods**:
For the copper rod:
\[
L_{cu} = L_{0,cu} (1 + \alpha_{cu} \Delta T)
\]
For the aluminum rod:
\[
L_{al} = L_{0,al} (1 + \alpha_{al} \Delta T)
\]
Given:
- \( L_{0,cu} = 88 \, \text{cm} \)
- \( \alpha_{cu} = 1.7 \times 10^{-5} \, \text{K}^{-1} \)
- \( \alpha_{al} = 2.2 \times 10^{-5} \, \text{K}^{-1} \)
3. **Set the Lengths Equal**:
Since the increase in length is independent of the increase in temperature, we can set the equations equal to each other:
\[
L_{al} - L_{cu} = 0
\]
This implies:
\[
L_{al} + L_{0,al} \alpha_{al} \Delta T = L_{0,cu} + L_{0,cu} \alpha_{cu} \Delta T
\]
4. **Rearranging the Equation**:
Rearranging gives us:
\[
L_{al} - L_{0,cu} = L_{0,cu} \alpha_{cu} \Delta T - L_{0,al} \alpha_{al} \Delta T
\]
Factoring out \( \Delta T \):
\[
L_{al} - 88 = \Delta T (88 \alpha_{cu} - L_{0,al} \alpha_{al})
\]
Since the lengths are independent of temperature, we can set the coefficients equal:
\[
L_{al} \alpha_{al} = L_{0,cu} \alpha_{cu}
\]
5. **Substituting Values**:
Substitute the known values into the equation:
\[
L_{al} \cdot 2.2 \times 10^{-5} = 88 \cdot 1.7 \times 10^{-5}
\]
6. **Solving for \( L_{al} \)**:
\[
L_{al} = \frac{88 \cdot 1.7 \times 10^{-5}}{2.2 \times 10^{-5}}
\]
Simplifying this gives:
\[
L_{al} = \frac{88 \cdot 1.7}{2.2}
\]
\[
L_{al} = \frac{149.6}{2.2} \approx 68 \, \text{cm}
\]
### Final Answer:
The length of the aluminum rod is approximately **68 cm**.
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