Topographical Contour Mapping & Road Grades

In civil and construction engineering, topographic surveys provide essential contour maps for highway grading, stormwater drainage channels, and earthwork cut-and-fill estimates. Highway safety regulations set by the Department of Transportation limit maximum allowable road grades (slopes) to ensure that loaded commercial haul trucks and transit vehicles can ascend without engine overheating and brake failure.

A civil engineering team is surveying a mountain valley where ground elevation $z$ (measured in dekameters, $1\text{ dam} = 10\text{ m}$) is modeled by the multivariable paraboloid function:

where $x$ and $y$ represent horizontal grid coordinates (in kilometers) measured east and north from a construction benchmark.

1. Determine the algebraic equations for the level curves (contour lines) at elevations $c = 1\text{ dam}$ ($10\text{ m}$) and $c = 4\text{ dam}$ ($40\text{ m}$). Describe their 2D geometric shape in the $xy$-plane and state their radii.
2. Determine the algebraic equation of the vertical cross-sectional trace along the cutting plane $y = 0$ (the $xz$-plane). Describe the 2D profile curve of this mountain slice.
3. Meaning & Real-Life Application: Explain what the spacing between successive elevation contours indicates about terrain steepness, and how civil engineers utilize vertical traces to verify that roadway cuts do not exceed legal grade percentage limits.