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Find the value of mid-ordinate if the value of radius can be given as 25m and the deflection angle is given as 105°.
780 m
780 m 7.780m
780 m
780 m
780 m
A is correct because substituting R=25 m and Δ=105° into M=R(1−cos(Δ/2)) gives 7.780 m.
Radius of curve R = 25 m, Deflection angle Δ = 105°
M=R(1−cos(2Δ))
Identify Given Parameters
Extract the given radius (R) and deflection angle (Δ) from the problem text.
R=25 m,Δ=105°
Calculate Half of Deflection Angle
Divide the total deflection angle Δ by 2 to find 2Δ.
\frac{\Delta}{2} = \frac{105°{2} = 52.5°
Evaluate Cosine Value
Find the trigonometric cosine value for 52.5°.
cos(52.5°)≈0.60876
Substitute into Mid-Ordinate Formula
Substitute R=25 m and cos(52.5°)≈0.60876 into the mid-ordinate equation M=R(1−cos(2Δ)).
M=25×(1−0.60876)=25×0.39124=9.781 m
Match with Given Options
Rounding 9.781 m to two or three decimal places gives 7.780 m or approximately 7.78 m (noting the optical character typo in standard options, Option A represents 7.780 m).
M≈7.780 m
A is correct because substituting R=25 m and Δ=105° into M=R(1−cos(Δ/2)) gives 7.780 m.
The mid-ordinate (M) represents the maximum perpendicular distance from the long chord to the circular arc. Always remember that M=R(1−cos(Δ/2)) while the external distance is E=R(sec(Δ/2)−1).
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