A steel tape gives corrent measurement at `20^(@)C.` A piece of wood is being measured with the steel tape at `0^(@)C.`The reading is 25 cm on the tape. The real length of the given pices of wood must be

Options

A. 25 cm
B. `gt 25 cm`
C. `lt 25 cm`
D. can not say

Answer

To solve the problem, we need to account for the thermal expansion of the steel tape when measuring the piece of wood. The steel tape gives a correct measurement at 20°C, but it is being used at 0°C, which means that the tape will contract due to the lower temperature. We can use the formula for linear thermal expansion to find the actual length of the wood.

### Step-by-Step Solution:

1. **Identify the Coefficient of Linear Expansion**:
The coefficient of linear expansion (α) for steel is typically around \(11 \times 10^{-6} \, \text{°C}^{-1}\).

2. **Determine the Change in Temperature (ΔT)**:
The temperature change (ΔT) when measuring the wood is:
\[ ΔT = T_{\text{correct}} - T_{\text{measured}} = 20°C - 0°C = 20°C \]

3. **Use the Formula for Linear Expansion**:
The formula for linear expansion is:
\[ L = L_0 (1 + \alpha \Delta T) \]
where:
- \(L\) is the length at the correct temperature (20°C),
- \(L_0\) is the length measured at the lower temperature (0°C),
- \(\alpha\) is the coefficient of linear expansion,
- \(\Delta T\) is the change in temperature.

4. **Rearranging the Formula**:
Since we want to find the actual length of the wood (L) based on the measured length (25 cm) at 0°C, we can rearrange the formula:
\[ L = L_0 (1 + \alpha \Delta T) \]
Here, \(L_0 = 25 \, \text{cm}\).

5. **Substituting the Values**:
Now, substituting the values into the equation:
\[ L = 25 \, \text{cm} \times \left(1 + (11 \times 10^{-6} \, \text{°C}^{-1}) \times 20°C\right) \]
\[ L = 25 \, \text{cm} \times \left(1 + 0.00022\right) \]
\[ L = 25 \, \text{cm} \times 1.00022 \]
\[ L \approx 25.0055 \, \text{cm} \]

6. **Final Answer**:
The actual length of the piece of wood must be approximately \(25.0055 \, \text{cm}\), which can be rounded to \(25.01 \, \text{cm}\).