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 understand how the measurement of the steel tape changes with temperature. The steel tape is calibrated to give accurate measurements at a specific temperature (20°C in this case). When the temperature changes, the length of the steel tape will also change due to thermal expansion.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Correct measurement temperature for steel tape: \( T_0 = 20°C \)
- Measurement temperature: \( T = 0°C \)
- Reading on the tape: \( L_{measured} = 25 \, \text{cm} \)

2. **Understand Thermal Expansion:**
- The length of the steel tape will contract when the temperature decreases. The formula for the change in length due to temperature change is:
\[ L = L_0 (1 + \alpha \Delta T) \]
- Where:
- \( L_0 \) is the original length at the reference temperature.
- \( \alpha \) is the coefficient of linear expansion for steel.
- \( \Delta T = T - T_0 \) is the change in temperature.

3. **Calculate the Change in Temperature:**
- The change in temperature \( \Delta T \) is:
\[ \Delta T = 0°C - 20°C = -20°C \]

4. **Apply the Thermal Expansion Formula:**
- Since the tape is measuring at a lower temperature, the actual length of the tape at 0°C can be expressed as:
\[ L_{actual} = L_{measured} (1 - \alpha \Delta T) \]
- Substituting the values:
\[ L_{actual} = 25 \, \text{cm} (1 - \alpha (-20)) \]
- This simplifies to:
\[ L_{actual} = 25 \, \text{cm} (1 + 20\alpha) \]

5. **Determine the Real Length of the Wood:**
- Since the steel tape is shorter at lower temperatures, the real length of the piece of wood must be:
\[ L_{real} = L_{measured} - \Delta L \]
- Where \( \Delta L = L_{measured} \cdot \alpha \cdot \Delta T \):
\[ \Delta L = 25 \cdot \alpha \cdot (-20) \]
- Therefore, the real length of the wood will be less than the measured length:
\[ L_{real} < 25 \, \text{cm} \]

6. **Conclusion:**
- The real length of the given piece of wood must be less than 25 cm.

### Final Answer:
The real length of the given piece of wood must be less than 25 cm.