Cubic parabolic transition curve for highways: select the standard relation (offset from tangent) commonly used for setting out.

Introduction / Context:
Transition curves provide a gradual change of curvature from tangent to circular arc. A widely used highway transition is the cubic parabola, chosen for its simple field equations and smooth curvature variation.

Given Data / Assumptions:

• x = distance measured along the tangent from the tangent point.
• y = offset from the tangent to the transition curve.
• R = radius of the circular curve that the transition leads into.
• L = length of the transition curve.

Concept / Approach:
For a cubic parabola, curvature varies linearly with arc length. The standard highway setting-out relation between the tangent offset y and station distance x is y = x^3 / (6 R L) (for small angles), enabling quick computation of offsets.

Step-by-Step Solution:

Verification / Alternative check:
Compare with spiral geometry tables; the cubic-parabola approximation yields close agreement for typical transition lengths, validating field use.

Why Other Options Are Wrong:
Option (a) is a parabolic form unrelated to cubic transition; (c) and (d) have incorrect dimensions and omit L or misplace R and L; thus they do not represent the standard highway relation.

Common Pitfalls:
Using large-angle geometry beyond the range where the approximation holds; mixing up coordinate systems (offset from chord vs tangent).

Final Answer:
y = x^3 / (6 R L)