To determine the change in length ([tex]\(\Delta L\)[/tex]) of the Golden Gate Bridge when it is exposed to the given temperature range, we will rely on the formula for linear expansion of materials. This formula is given by:
[tex]\[
\Delta L = \alpha \cdot L_0 \cdot \Delta T
\][/tex]
where:
- [tex]\(\Delta L\)[/tex] is the change in length,
- [tex]\(\alpha\)[/tex] is the coefficient of linear expansion (for steel, [tex]\(\alpha = 12 \times 10^{-6} \frac{1}{\text{°C}}\)[/tex]),
- [tex]\(L_0\)[/tex] is the original length (1275 meters in this case),
- [tex]\(\Delta T\)[/tex] is the change in temperature.
### Step-by-Step Solution:
1. Identify the given values:
- Original length ([tex]\(L_0\)[/tex]): 1275 meters
- Lowest temperature ([tex]\(T_{\text{low}}\)[/tex]): [tex]\(-18^{\circ}C\)[/tex]
- Highest temperature ([tex]\(T_{\text{high}}\)[/tex]): [tex]\(39^{\circ}C\)[/tex]
- Coefficient of linear expansion for steel ([tex]\(\alpha\)[/tex]): [tex]\(12 \times 10^{-6} \frac{1}{\text{°C}}\)[/tex]
2. Calculate the change in temperature ([tex]\(\Delta T\)[/tex]):
[tex]\[
\Delta T = T_{\text{high}} - T_{\text{low}}
\][/tex]
[tex]\[
\Delta T = 39 - (-18)
\][/tex]
[tex]\[
\Delta T = 39 + 18
\][/tex]
[tex]\[
\Delta T = 57^{\circ}C
\][/tex]
3. Use the linear expansion formula to find [tex]\(\Delta L\)[/tex]:
[tex]\[
\Delta L = \alpha \cdot L_0 \cdot \Delta T
\][/tex]
[tex]\[
\Delta L = (12 \times 10^{-6}) \cdot 1275 \cdot 57
\][/tex]
4. Calculate the product:
[tex]\[
\Delta L = 12 \times 10^{-6} \cdot 1275 \cdot 57
\][/tex]
[tex]\[
\Delta L = 0.8721 \text{ meters}
\][/tex]
Therefore, the change in length of the Golden Gate Bridge between the temperatures of [tex]\(-18^{\circ}C\)[/tex] and [tex]\(39^{\circ}C\)[/tex] is [tex]\(0.8721\)[/tex] meters.
[tex]\[
\boxed{\Delta L = 0.8721 \text{ meters}}
\][/tex]
