A steel beam being used in the construction of a skyscraper has a length of 39.000 m when delivered on a cold day at a temperature of 16.000°F. What is the length of the beam when it is being installed later on a warm day when the temperature is 91.000°F? (Give your answer to at least five significant figures.)

To determine the change in length of the steel beam due to temperature change, we can use the formula for linear thermal expansion:

\[ \Delta L = L_0 \alpha \Delta T \]

where:
- \(\Delta L\) is the change in length,
- \(L_0\) is the original length of the beam,
- \(\alpha\) is the coefficient of linear expansion for steel,
- \(\Delta T\) is the change in temperature.

Let's start by converting the temperatures from Fahrenheit to Celsius, since the coefficient of linear expansion is typically given in \(\text{°C}^{-1}\).

The formula to convert Fahrenheit to Celsius is:
\[ T(°C) = \frac{5}{9} (T(°F) - 32) \]

For the initial temperature:
\[ T_i(°C) = \frac{5}{9} (16 - 32) = \frac{5}{9} (-16) = -\frac{80}{9} ≈ -8.8889 \text{°C} \]

For the final temperature:
\[ T_f(°C) = \frac{5}{9} (91 - 32) = \frac{5}{9} (59) = \frac{295}{9} ≈ 32.7778 \text{°C} \]

Next, calculate the temperature change:
\[ \Delta T = T_f - T_i = 32.7778 - (-8.8889) = 32.7778 + 8.8889 = 41.6667 \text{°C} \]

The coefficient of linear expansion (\(\alpha\)) for steel is approximately \(11 \times 10^{-6} \text{°C}^{-1}\).

Now, we can calculate the change in length (\(\Delta L\)):
\[ \Delta L = L_0 \alpha \Delta T \]
\[ \Delta L = 39.000 \times 11 \times 10^{-6} \times 41.6667 \]
\[ \Delta L ≈ 39.000 \times 11 \times 41.6667 \times 10^{-6} \]
\[ \Delta L ≈ 39.000 \times 458.3337 \times 10^{-6} \]
\[ \Delta L ≈ 17.874 \times 10^{-3} \]
\[ \Delta L ≈ 0.017874 \text{ m} \]

Finally, the new length of the beam when installed at the warmer temperature is:
\[ L_f = L_0 + \Delta L \]
\[ L_f = 39.000 + 0.017874 \]
\[ L_f ≈ 39.017874 \text{ m} \]

So, the length of the beam when it is being installed later on a warm day at 91.000°F is approximately 39.017874 meters.

Would you like more details or have any questions?

Here are some related questions:
1. What is the formula for linear thermal expansion?
2. How do you convert temperatures from Fahrenheit to Celsius?
3. What is the coefficient of linear expansion and how is it used?
4. Why do materials expand when heated?
5. How does the coefficient of linear expansion vary among different materials?
6. What are some practical applications of thermal expansion in construction?
7. Can thermal expansion cause structural issues in buildings?
8. How do engineers account for thermal expansion in design?

**Tip:** Always consider the material properties and environmental conditions when designing structures to ensure stability and safety.

Learn how to calculate the length change of a steel beam due to temperature from 16.000°F to 91.000°F. This problem covers linear thermal expansion using the coefficient of linear expansion for steel. Suitable for advanced high school and college-level physics and engineering students.

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