Highway Interchange Overpass Alignment (Civil Engineering)

In transportation and highway civil engineering, grade elevation transitions for elevated freeway interchange ramps must provide smooth vertical curves that merge seamlessly into constant descending highway grades. Abrupt angular transitions cause violent vertical accelerations for high-speed traffic, while incorrect asymptotic planning leads to poor storm drainage runoff and hazardous ponding on overpass bridge decks.

A civil engineering team at the Nevada Department of Transportation (NDOT) is designing the profile elevation $y(x)$ (in meters above sea level) of an interstate connector ramp as a function of horizontal station distance $x$ (in hundreds of meters, with station $x \ge 0$). The elevation profile is modeled by the rational function:

1. Determine the domain and vertical asymptote of the rational elevation profile for all real $x$, and explain why the vertical asymptote falls outside the physical roadway domain.
2. Calculate the vertical ramp elevation at the start station $x = 0$ ($y$-intercept).
3. Use the method of polynomial long division to express $y(x)$ in quotient-remainder form, identifying the equation of the slant (oblique) asymptote $y = mx + b$.
4. Interpret the physical engineering meaning of the slope $m$ and intercept $b$ of the slant asymptote in the context of long-range highway grading.
5. Determine whether the bridge deck profile approaches the slant asymptote grade line from above or below as horizontal stationing distance $x$ extends toward infinity.