Suppose a railroad rail is 6 kilometers and it expands on a hot day by 15 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 
  
21.2 meters above the ground.
​(Type an integer or decimal rounded to the nearest tenth as​ needed.)

This problem involves an application of the concept of thermal expansion, which can be approximated using geometry.

When the railroad rail expands due to heat, it forms an arc. We assume that the rail remains flat and forms a circular segment. The expansion creates a small rise at the center of the rail, and we need to calculate the height of this rise.

### Given:
- Original length of the rail (chord length): \( L = 6000 \) meters
- Expansion of the rail: 15 cm = 0.15 meters
- New length of the rail: \( L' = 6000 + 0.15 = 6000.15 \) meters

We assume the rail forms a part of a circle. The center of the rail will rise above the ground, and we want to find that height (denoted as \( h \)).

### Steps:

1. The new length \( L' \) represents the arc length of the circular segment.
2. The chord length \( L \) remains 6000 meters.
3. The height \( h \) represents the maximum distance the rail rises above the chord.

To solve this, we can use an approximation for small expansions, where \( h \) (the rise in height) is given by:

\[
h \approx \sqrt{\frac{L' \cdot (L' - L)}{2}}
\]

Substituting the values \( L = 6000 \) meters and \( L' = 6000.15 \) meters into this equation:

\[
h \approx \sqrt{\frac{6000.15 \cdot (6000.15 - 6000)}{2}}
\]

Let me calculate this for you.The center of the rail would rise approximately 21.2 meters above the ground.

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:
1. How would the rise change if the rail expanded by 30 centimeters instead?
2. What is the relationship between thermal expansion and temperature increase?
3. How does the material of the rail affect the expansion rate?
4. Can this formula be applied to longer or shorter railroad tracks with similar expansions?
5. How would the rise differ if the track were curved instead of straight?

**Tip:** Thermal expansion coefficients for different materials are crucial when designing structures that experience temperature changes.

Suppose a railroad rail is 6 kilometers and it expands on a hot day by 15 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 21.2 meters above the ground. (Type an integer or decimal rounded to the nearest tenth as needed.)

Learn how to solve a thermal expansion problem where a railroad rail expands by 15 cm over a length of 6 kilometers. Using geometric concepts, we calculate that the rail would rise approximately 21.2 meters above the ground. Suitable for students in Grades 10-12.

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