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We are given a problem where the length \( L \) of a steel bridge depends on the temperature \( H \), which in turn depends on time \( t \). Specifically, we know:

1. The length of the bridge increases by 0.5 μm for every 1°C increase in temperature.
2. The temperature \( H \) is increasing at a rate of 6°C per hour.

We are asked to find the rate at which the length of the bridge is increasing in micrometers per hour (μm/hour).

### Step-by-step solution:

1. **Relating \( L \) to \( H \):**
   We know that \( L \) increases by 0.5 μm for every degree increase in temperature \( H \), so we can write:
   \[
   \frac{dL}{dH} = 0.5 \, \text{μm/°C}
   \]
   
2. **Relating \( H \) to \( t \):**
   We are told that the temperature \( H \) is increasing at 6°C per hour, so:
   \[
   \frac{dH}{dt} = 6 \, \text{°C/hour}
   \]

3. **Using the chain rule to find \( \frac{dL}{dt} \):**
   We want to find the rate at which the length \( L \) is increasing with respect to time \( t \), or \( \frac{dL}{dt} \). Using the chain rule:
   \[
   \frac{dL}{dt} = \frac{dL}{dH} \times \frac{dH}{dt}
   \]
   Substituting the given values:
   \[
   \frac{dL}{dt} = (0.5 \, \text{μm/°C}) \times (6 \, \text{°C/hour})
   \]

4. **Calculating the result:**
   \[
   \frac{dL}{dt} = 3 \, \text{μm/hour}
   \]

### Final answer:
The length of the bridge is increasing at a rate of **3 μm/hour**.

Would you like more details on any part of this solution, or have any other questions?

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Here are 5 related questions to further explore this concept:
1. How would the rate of change of the length change if the temperature increased by 10°C per hour instead of 6°C?
2. If the steel bridge cools down, how would you adjust the calculation for decreasing temperatures?
3. Can you derive the equation for the total length of the bridge after a given time if the initial length and temperature are known?
4. How does the material's coefficient of thermal expansion affect the relationship between length and temperature?
5. What happens if the length increase is nonlinear with respect to temperature?

**Tip:** Always pay attention to units when dealing with rates of change—this ensures consistent calculations across different physical quantities.

The length, L, in micrometers (μm), of steel depends on the air temperature H in degrees celsius, and the temperature H depends on time, t, measured in hours. If the length of a steel bridge increases by 0.5 μm for every degree increase in temperature, and the temperature is increasing at 6 degrees celsius per hour, how fast is the length of the bridge increasing?

Learn how to calculate the rate of change in the length of a steel bridge as the temperature increases by 6°C per hour. This problem involves key calculus concepts, including the chain rule and rates of change, and is suitable for students in Grades 10-12.

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