To find the true length of the measured line, we account for the thermal expansion of the steel tape. The formula for linear expansion is:

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

Where:

• $\Delta L$ = change in length of the tape due to temperature change.
• $L_0$ = original length of the tape at the reference temperature ($55 \, \text{m}$ at $30^\circ \, \text{C}$).
• $\alpha$ = coefficient of linear expansion for steel ($0.0000116 \, \text{per} \, \degree \text{C}$).
• $\Delta T$ = temperature change ($45^\circ \, \text{C} - 30^\circ \, \text{C} = 15^\circ \, \text{C}$).

The expansion of the tape is: $\Delta L = 55 \cdot 0.0000116 \cdot 15 = 0.00957 \, \text{m}.$

Thus, the expanded length of the tape is: $L_{\text{new}} = L_0 + \Delta L = 55 + 0.00957 = 55.00957 \, \text{m}.$

The tape measured a line as $532.28 \, \text{m}$. To find the true length of the line ($L_{\text{true}}$), we scale it by the ratio of the actual tape length to the nominal tape length:

$L_{\text{true}} = \frac{L_{\text{measured}} \cdot L_{\text{new}}}{L_0}.$

Substituting the values: $L_{\text{true}} = \frac{532.28 \cdot 55.00957}{55}.$

Simplify: $L_{\text{true}} = 532.37296 \, \text{m}.$

Final Answer:

The true length of the line is approximately $\mathbf{532.373 \, \text{m}}$.

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Related Questions:

1. How does the coefficient of linear expansion affect the true length calculation?
2. What would happen if the tape was used at a lower temperature than $30^\circ \, \text{C}$?
3. How is the formula for linear expansion derived?
4. What happens if the coefficient of linear expansion is doubled?
5. How would you calculate the new length if the tape was $100 \, \text{m}$ long instead?

Tip: Always verify the reference temperature and the coefficient of linear expansion when dealing with thermal expansion problems!