To determine how much taller the Eiffel Tower becomes when the temperature increases by 15°C, we need to use the concept of linear expansion. The formula for linear expansion is:
\[ \Delta L = \alpha L_0 \Delta T \]
Where:
- \( \Delta L \) is the change in length (or height in this case).
- \( \alpha \) is the coefficient of linear expansion for the material.
- \( L_0 \) is the original length (or height) of the material.
- \( \Delta T \) is the change in temperature.
Let's plug the values into the formula:
Given:
- The original height of the Eiffel Tower, \( L_0 \), is 321 m.
- The temperature increase, \( \Delta T \), is 15°C.
- The coefficient of linear expansion for steel, \( \alpha \), is \( 11 \times 10^{-6} \ / ^\circ C \).
Now calculate the increase in height \( \Delta L \):
\[ \Delta L = (11 \times 10^{-6} \ / ^\circ C) \times 321 \ m \times 15 ^\circ C \]
\[ \Delta L = (11 \times 10^{-6}) \times 321 \times 15 \]
\[ \Delta L = 11 \times 321 \times 15 \times 10^{-6} \]
\[ \Delta L = 52965 \times 10^{-6} \]
\[ \Delta L = 0.052965 \ m \]
To convert the increase in height from meters to centimeters, we multiply by 100 (since 1 m = 100 cm):
\[ \Delta L_{cm} = 0.052965 \ m \times 100 \ cm/m \]
\[ \Delta L_{cm} = 5.2965 \ cm \]
Now we round to the nearest whole number:
\[ \Delta L_{cm} \approx 5 \ cm \]
The Eiffel Tower becomes approximately 5 cm taller when the temperature increases by 15°C.
