5. suppose a railroad is 2 km long, and it expands on a hot day by 50 cm in length. approximately how high would the center of the rail rise above the ground? (hint: convert all measurements to meters before calculating any values)

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

1.9 m