The cross-section of a railway tunnel is shown. The length of the base $AB$ is 100 m, and the radius of the arc is 56 m. We need to calculate the perimeter of the cross-section to the nearest metre.

1. Problem Description

The cross-section of a railway tunnel is shown. The length of the base $AB$ is 100 m, and the radius of the arc is 56 m. We need to calculate the perimeter of the cross-section to the nearest metre.

2. Solution Steps

The perimeter of the cross-section is the sum of the length of the line segment $AB$ and the length of the arc. The length of $AB$ is given as 100 m.

Since the arc is a semicircle, its length is half the circumference of a circle with radius 56 m.

The formula for the circumference of a circle is:

$C = 2 \pi r$

The length of the semicircle is therefore:

$L = \frac{1}{2} (2 \pi r) = \pi r$

Given that $r = 56$ m, the length of the arc is:

$L = \pi (56)$

Using the value $\pi \approx 3.14159$, we get:

$L \approx 3.14159 \times 56 \approx 175.929$ m.

The perimeter of the cross-section is the sum of the length of $AB$ and the length of the arc:

$P = AB + L \approx 100 + 175.929 = 275.929$ m.

Rounding to the nearest metre, we get 276 m.

3. Final Answer

The perimeter of the cross-section is approximately 276 m.