Suppose a railroad rail is 2 kilometers and it expands on a hot day by 19 centimeters in length. Approximately how many meters would the center of the rail rise above the​ ground? 
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Part 1
The center of the rail would rise approximately 
  
enter your response here meters above the ground.
​(Type an integer or decimal rounded to the nearest tenth as​ needed.)

We can model the situation by considering the rail as a segment of a circle with a small rise in the middle. Let’s use geometry to estimate how high the rail will rise due to the expansion.

### Given:
- Initial length of the rail \( L_0 = 2000 \, \text{m} \) (since 2 kilometers = 2000 meters)
- Expansion of the rail \( \Delta L = 0.19 \, \text{m} \) (since 19 centimeters = 0.19 meters)

Now, let's consider the rail as forming a circular arc after expanding, with the chord being the original length \( L_0 \), and the arc length being the expanded length \( L = L_0 + \Delta L \).

### Step 1: Find the radius of the circle.
The relationship between the arc length \( L \), the chord length \( L_0 \), and the radius \( R \) is approximated by:
\[
L = R \theta
\]
where \( \theta \) is the central angle in radians.

For small expansions, the angle \( \theta \) can be approximated as:
\[
\theta \approx \sqrt{\frac{6 \Delta L}{L_0}}
\]
Substituting the given values:
\[
\theta \approx \sqrt{\frac{6 \times 0.19}{2000}} = \sqrt{0.00057}
\]
\[
\theta \approx 0.02385 \, \text{radians}
\]

### Step 2: Calculate the rise of the center.
The height \( h \) (the rise of the center of the rail) is related to the radius and angle by:
\[
h \approx \frac{L_0}{2} \sin\left(\frac{\theta}{2}\right)
\]
For small angles, \( \sin\left(\frac{\theta}{2}\right) \approx \frac{\theta}{2} \). Thus:
\[
h \approx \frac{L_0}{2} \times \frac{\theta}{2} = 1000 \times \frac{0.02385}{2}
\]
\[
h \approx 1000 \times 0.01193 = 11.93 \, \text{meters}
\]

### Final Answer:
The center of the rail would rise approximately **11.9 meters** above the ground.

Would you like further details or have any questions?

Here are five related questions to think about:
1. How would the rise change if the rail expanded by 25 centimeters instead of 19?
2. What is the relationship between the expansion length and the height of the rise?
3. How would the result differ for a 5-kilometer-long rail?
4. What other factors, besides temperature, might affect rail expansion?
5. How does the geometry of arcs help in calculating similar problems?

**Tip**: When working with small angles, approximations like \( \sin \theta \approx \theta \) for small \( \theta \) can simplify calculations without much loss in accuracy.

Suppose a railroad rail is 2 kilometers and it expands on a hot day by 19 centimeters in length. Approximately how many meters would the center of the rail rise above the ground? The center of the rail would rise approximately ___ meters above the ground. (Type an integer or decimal rounded to the nearest tenth as needed.)

Explore how to calculate the rise of a railroad rail due to thermal expansion. This problem involves key concepts of geometry and trigonometry, including circle arcs, small angle approximation, and arc length formulas. Ideal for students in Grades 10-12.

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