To calculate the true distance between the two points, we need to adjust the measured length for the temperature change. This is done using the coefficient of linear expansion for steel, which tells us how much a material expands or contracts with temperature.

Formula for length change due to temperature:

$\Delta L = L_0 \cdot \alpha \cdot \Delta T$

Where:

• $\Delta L$ is the change in length,
• $L_0$ is the original length at the reference temperature (which is 30 m in this case),
• $\alpha$ is the coefficient of linear expansion for steel ($\alpha = 12 \times 10^{-6} \, \text{°C}^{-1}$),
• $\Delta T = T_{\text{final}} - T_{\text{initial}}$ is the change in temperature.

Given data:

• $L_0 = 25.970 \, \text{m}$ (measured length at 5.00°C),
• $T_{\text{initial}} = 20.0 \, \text{°C}$ (temperature at which the tape is accurate),
• $T_{\text{final}} = 5.00 \, \text{°C}$,
• $\alpha = 12 \times 10^{-6} \, \text{°C}^{-1}$.

Step 1: Calculate the temperature change

$\Delta T = T_{\text{final}} - T_{\text{initial}} = 5.00 - 20.0 = -15.0 \, \text{°C}$

Step 2: Calculate the length change

$\Delta L = L_0 \cdot \alpha \cdot \Delta T$

$\Delta L = 25.970 \cdot (12 \times 10^{-6}) \cdot (-15.0)$

$\Delta L = -0.004676 \, \text{m}$

This means the tape contracted by 0.004676 meters due to the temperature decrease.

Step 3: Calculate the true distance

The measured length was too short due to the contraction of the steel tape. To find the true length, we add the length change:

$\text{True length} = L_{\text{measured}} + \Delta L$

$\text{True length} = 25.970 \, \text{m} + 0.004676 \, \text{m}$

$\text{True length} = 25.9747 \, \text{m}$

So, the true distance between the two points is approximately 25.975 meters.

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Would you like more details on any part of this solution, or have any questions?

Related Questions:

1. What would happen if the temperature increased instead of decreased?
2. How can we calculate the coefficient of linear expansion for a material experimentally?
3. What is the effect of different materials on the accuracy of measurements with a steel tape?
4. How does temperature affect other types of measurements, such as volume or area?
5. What would be the effect of using a tape made of a different material, such as nylon, in this situation?

Tip:

Always ensure the temperature of your measuring instruments is taken into account when accuracy is crucial, especially in long-distance measurements!