Suppose a railroad rail is 5 kilometers and it expands on a hot day by 18 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.)

This problem can be solved by assuming that the rail, after expanding due to heat, forms an arc. The total length of the rail increases, and the arc formed resembles a circle segment. We are looking for the height (h) of the arc formed at the center, given the expansion and original length of the rail.

### Known values:
- Original length of the rail = 5 kilometers = 5000 meters
- Expansion in length = 18 centimeters = 0.18 meters

### Approach:

1. **Arc length formula**: 
   The expanded length of the rail will be the arc length \( L \), which is:
   \[
   L = \text{original length} + \text{expansion}
   = 5000 + 0.18 = 5000.18 \, \text{meters}.
   \]

2. **Simplified geometric model**: 
   We approximate the rail as forming a very flat circle segment where the total arc length is \( L = 5000.18 \) meters, and the original distance (the chord length) remains 5000 meters. We aim to find the height (h) at the center of this segment.

We can use the formula for the height \( h \) of a circular arc when the arc length and chord length are known:

\[
h = \sqrt{ \left(\frac{L}{2}\right)^2 - \left(\frac{chord}{2}\right)^2}
\]

Now let’s compute this.To proceed manually:

Given the formula:

\[
h = \sqrt{ \left(\frac{L}{2}\right)^2 - \left(\frac{chord}{2}\right)^2 }
\]

1. Half of the expanded length (arc length) \( \frac{L}{2} = \frac{5000.18}{2} = 2500.09 \, \text{m} \).
2. Half of the original length (chord) \( \frac{chord}{2} = \frac{5000}{2} = 2500 \, \text{m} \).

Now plug these into the equation:

\[
h = \sqrt{(2500.09)^2 - (2500)^2}
\]

\[
h \approx \sqrt{6250450.0081 - 6250000}
\]

\[
h \approx \sqrt{450.0081} \approx 21.2 \, \text{meters}
\]

Thus, the center of the rail would rise approximately **21.2 meters** above the ground.

Would you like further details on this? 

### Related Questions:
1. What is the relationship between chord length and arc height in a circular arc?
2. How does the length of expansion affect the height of the arc?
3. How would a larger expansion, such as 50 cm, affect the rise in height?
4. What assumptions are made in approximating the rail as a circular arc?
5. How does temperature affect the expansion of metals like steel in real-life scenarios?

### Tip:
Always ensure that units are consistent when working with geometric or physical problems. Converting all values to the same unit helps avoid calculation errors.

Suppose a railroad rail is 5 kilometers and it expands on a hot day by 18 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.)

Solve a real-world problem where a railroad rail expands by 18 cm over 5 kilometers. Learn how to calculate the height of the arc formed using geometric principles, including arc length and chord length formulas. This detailed solution is perfect for students in Grades 10-12.

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