The original length of the railroad is
L₁ = 2 km = 2000 km
The extended length after expansion is
L₂ = L₁ + 50 cm
= 2000 + 0.5
= 2000.5 m
Assume that the deflected shape is a circle with radius = r, as shown in the figure below.
The central angle of the deflected shape is 2θ.
The deflected length is calculated as
2rθ = L₂.
That is
rθ = 2000.5/2 = 1000.25
r = 1000.25/θ (1)
By definition (from the figure)
sinθ = 1000/r (2)
Substitute (1) into (2).
sin θ = (1000 θ)/1000.25 = 0.99975 θ
To find θ, define the function
f(θ) = 0.99975 θ - sin θ
A graphical solution from the calculator yields
θ = 0.0038 rad.
Therefore from (1), obtain
r = 263223.7 m
The height of the center of the rail above ground is
h = r - r cos θ = r(1 - cos θ)
= 263223.7(1 - cos(0.0038))
= 1.9 m
Answer: 1.9 m
