Answer:
To solve this problem, we can use the coefficient of linear expansion for aluminum, which is approximately \( \alpha = 23.1 \times 10^{-6} \, \text{°C}^{-1} \). The formula for linear expansion is:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]
Where:
- \( \Delta L \) is the change in length
- \( L_0 \) is the original length
- \( \alpha \) is the coefficient of linear expansion
- \( \Delta T \) is the change in temperature
Given that \( L_0 = 40 \, \text{m} \), \( \Delta L = 0.08 \, \text{m} \), and \( \Delta T \) is what we need to find. We rearrange the formula to solve for \( \Delta T \):
\[ \Delta T = \frac{\Delta L}{L_0 \cdot \alpha} \]
Now, plug in the given values:
\[ \Delta T = \frac{0.08 \, \text{m}}{40 \, \text{m} \cdot (23.1 \times 10^{-6} \, \text{°C}^{-1})} \]
\[ \Delta T \approx \frac{0.08}{40 \times 23.1 \times 10^{-6}} \, \text{°C} \]
\[ \Delta T \approx \frac{0.08}{0.000924} \, \text{°C} \]
\[ \Delta T \approx 86.58 \, \text{°C} \]
Now, to find the temperature at which the wing would be 8 cm shorter, we add this change in temperature to the original temperature:
\[ \text{New temperature} = 21°C + 86.58°C \]
\[ \text{New temperature} \approx 107.58°C \]
So, at approximately \( 107.58°C \), the wing would be \( 0.08 \, \text{m} \) shorter.
