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

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)

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2016-04-23 01:58:37

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