Suppose a railroad rail is 2 kilometers and it expands on a hot day by 20 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.

This problem can be approached as a geometric approximation, considering that the rail, when expanding, forms a very shallow arc, resembling part of a circle.

### Assumptions and Approach:
1. The original length of the rail is 2 km = 2000 meters.
2. The rail expands by 20 cm = 0.2 meters.
3. The new length of the rail is \( L' = 2000 + 0.2 = 2000.2 \, \text{meters} \).
4. We assume the ends of the rail are fixed, and it forms a shallow arc with a small rise at the center.

### Formula:
The rise of the arc \( h \) can be approximated using the formula for the relationship between the arc length, radius, and chord length for a small arc:

\[
L' \approx L + \frac{h^2}{2L}
\]

Where:
- \( L' \) is the expanded length of the rail.
- \( L \) is the original length of the rail.
- \( h \) is the rise at the center of the arc.

Let's solve for \( h \).

\[
2000.2 \approx 2000 + \frac{h^2}{2 \cdot 2000}
\]
\[
0.2 \approx \frac{h^2}{4000}
\]
\[
h^2 \approx 0.2 \times 4000 = 800
\]
\[
h \approx \sqrt{800} \approx 28.28 \, \text{meters}
\]

So, the center of the rail would rise approximately **28.28 meters** above the ground.

Let me know if you want further details or clarifications!

### Related Questions:
1. What assumptions are made in this type of approximation?
2. How does the rise change if the expansion is doubled?
3. How would the rise be affected if the original rail length were shorter?
4. What would happen if the rail were not fixed at both ends?
5. How does thermal expansion vary with temperature changes?

### Tip:
Thermal expansion is often linear with temperature changes, but real-life constraints like material properties and mounting conditions can alter results.

Suppose a railroad rail is 2 kilometers and it expands on a hot day by 20 centimeters in length. Approximately how many meters would the center of the rail rise above the ground?

Discover how to calculate the rise in the center of a railroad rail that expands by 20 centimeters over a 2-kilometer length. This problem involves concepts of thermal expansion and geometric approximation, making it ideal for high school students in Grades 10-12.

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