The problem involves a railroad rail expanding due to heat. When the rail expands, it no longer remains flat, forming an arc. We can treat this as an isosceles triangle, where the rail becomes the base of the triangle, and the height of the arc represents how much the center of the rail rises.

We need to calculate this height of the arc. This can be approximated using a geometric method.

Given dataThe problem involves a railroad rail expanding due to heat. When the rail expands, it no longer remains flat, forming an arc. We can treat this as an isosceles triangle, where the rail becomes the base of the triangle, and the height of the arc represents how much the center of the rail rises.

We need to calculate this height of the arc. This can be approximated using a geometric method.

Given

• The original length of the rail is $3 \, \text{km} = 3000 \, \text{m}$.
• The expansion in length is $20 \, \text{cm} = 0.2 \, \text{m}$.

The expanded length of the rail is now $3000.2 \, \text{m}$, which is longer than the original 3000 m.

Step 1: Approximation using a geometric relationship

We approximate the rise using the formula for the height of an arc when the expansion is small: $h \approx \sqrt{\frac{L^2 - S^2}{4}}$ Where:

• $h$ is the rise (height at the center),
• $L = 3000.2 \, \text{m}$ is the expanded length of the rail,
• $S = 3000 \, \text{m}$ is the original length.

Step 2: Plug in the values and calculate

$h \approx \sqrt{\frac{(3000.2)^2 - 3000^2}{4}}$

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Meanwhile, here are some related questions you might find useful:

1. How does expansion of materials affect engineering structures?
2. What is the formula for arc length, and how is it used?
3. How does thermal expansion vary with different materials?
4. Can thermal expansion be controlled or prevented in railroads?
5. What other real-world scenarios involve thermal expansion?

Tip: When working with thermal expansion problems, always pay attention to the units used for lengths and temperatures to avoid errors.