(a) An alumunium measuring rod which is correct at `5^@ C`, measures a certain distance as `88.42 cm at 35 ^@ C`. determine the error in measuring hr distance due to the expansion of the rod. (b) If this aluminium rod measures a length of steel as `88.42 cm at 35 ^@ C`, what is the correct length of the steel at `35 ^@ C` ?

Explanation

To solve the problem, we will break it down into two parts as given in the question.

### Part (a): Determine the error in measuring the distance due to the expansion of the rod.

1. **Identify the given values:**
- Length measured by the aluminum rod, \( L = 88.42 \, \text{cm} \)
- Initial temperature, \( T_1 = 5^\circ C \)
- Final temperature, \( T_2 = 35^\circ C \)
- Coefficient of linear expansion for aluminum, \( \alpha = 2.4 \times 10^{-5} \, \text{°C}^{-1} \)

2. **Calculate the change in temperature:**
\[ \Delta T = T_2 - T_1 = 35 - 5 = 30 \, \text{°C} \]

3. **Use the formula for linear expansion to find the error:**
\[ \Delta L = L \cdot \alpha \cdot \Delta T \]
Substituting the values:
\[ \Delta L = 88.42 \, \text{cm} \cdot (2.4 \times 10^{-5} \, \text{°C}^{-1}) \cdot 30 \, \text{°C} \]

4. **Calculate the error:**
\[ \Delta L = 88.42 \cdot 2.4 \times 10^{-5} \cdot 30 \]
\[ \Delta L = 88.42 \cdot 7.2 \times 10^{-4} = 0.0637 \, \text{cm} \approx 0.064 \, \text{cm} \]

Thus, the error in measuring the distance due to the expansion of the rod is approximately **0.064 cm**.

### Part (b): Determine the correct length of the steel at \( 35^\circ C \).

1. **Identify the measured length of steel:**
- Measured length of steel = \( 88.42 \, \text{cm} \) (as measured by the aluminum rod)

2. **Correct the measured length by adding the error:**
\[ L' = L + \Delta L \]
Substituting the values:
\[ L' = 88.42 \, \text{cm} + 0.064 \, \text{cm} \]

3. **Calculate the correct length of steel:**
\[ L' = 88.42 + 0.064 = 88.484 \, \text{cm} \]

Thus, the correct length of the steel at \( 35^\circ C \) is **88.484 cm**.

### Summary of Solutions:
- (a) The error in measuring the distance due to the expansion of the rod is **0.064 cm**.
- (b) The correct length of the steel at \( 35^\circ C \) is **88.484 cm**.