An aluminium measuring rod, which is correct at `5^@C` measures the length of a line as 80 cm at `45^@C` If thermal coefficient of linear expansion of aluminium is `2.50xx10^(-4)//^@C`, the correct length of the line is:

Options

A. A) `80.05cm`
B. B) `79.92cm`
C. C) `81.12cm`
D. D) `79.62cm`

Answer

To find the correct length of the line measured with an aluminium measuring rod at a different temperature, we can use the formula for linear expansion:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length measured at 45°C, \( L_t = 80 \, \text{cm} \)
- Correct temperature, \( T_0 = 5 \, \text{°C} \)
- Thermal coefficient of linear expansion, \( \alpha = 2.50 \times 10^{-4} \, \text{°C}^{-1} \)
- Change in temperature, \( \Delta T = T - T_0 = 45 - 5 = 40 \, \text{°C} \)

2. **Use the Linear Expansion Formula:**
The formula for linear expansion is given by:
\[ L_t = L_0 (1 + \alpha \Delta T) \]
where \( L_0 \) is the original length at the reference temperature.

3. **Rearranging the Formula:**
To find the correct length \( L_0 \), we rearrange the formula:
\[ L_0 = \frac{L_t}{1 + \alpha \Delta T} \]

4. **Calculate \( 1 + \alpha \Delta T \):**
Substitute the values into the equation:
\[ 1 + \alpha \Delta T = 1 + (2.50 \times 10^{-4}) \times 40 \]
\[ = 1 + 0.01 = 1.01 \]

5. **Substitute Back to Find \( L_0 \):**
Now substitute \( L_t \) and the calculated value into the rearranged formula:
\[ L_0 = \frac{80}{1.01} \]

6. **Perform the Calculation:**
\[ L_0 \approx 79.2079 \, \text{cm} \]

7. **Final Answer:**
Rounding to two decimal places, the correct length of the line is:
\[ L_0 \approx 79.21 \, \text{cm} \]