Perimeter of a Railway Tunnel Cross-Section
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3. The diagram shows the cross-section of a railway tunnel. If $|AB| = 100\,m$ and the radius of the arc is $56\,m$, calculate, correct to the nearest metre, the perimeter of the cross-section.
This question includes visual content: A diagram of a railway tunnel cross-section. It consists of a straight horizontal line segment AB at the bottom and a large circular arc connecting points A and B above it. The line segment AB is labeled with a length of 100m. The shape resembles a circle with a small portion (a segment) removed from the bottom.
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In this problem, we need to calculate the perimeter of a railway tunnel cross-section. The cross-section consists of a circular arc and a straight floor line connecting points A and B.
Perimeter of Tunnel Cross-section
From the given information, distance A B is one hundred meters. The radius of the circular arc is fifty-six meters. The total perimeter is the sum of the floor width and the major arc length.
To find the length of the arc, we first need to determine the central angle subtended by the chord A B. Let's visualize the geometry. Let O be the center of the circle.
We can split the isosceles triangle O A B into two right-angled triangles by dropping a perpendicular from O to the midpoint of A B.
Let theta be half of the central angle A O B. Using sine, we have sine of theta equals the opposite side, which is fifty, divided by the hypotenuse, which is fifty-six.
Solving for theta, we take the inverse sine of zero point eight nine two nine.
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