Topographical Contour Mapping & Road Grades Solution

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:

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.

Step-by-Step Solution:

1. Calculate Level Curves (Horizontal Slices): State the Level Curve Equation for Two-Variable Functions:

   $$f(x, y) = c$$
   • For $c = 1\text{ dam}$ ($10\text{ m}$):
     $$x^2 + y^2 = 1$$
     This is a circle in the $xy$-plane centered at the origin $(0, 0)$ with radius $r_1 = \sqrt{1} = 1\text{ km}$.
   • For $c = 4\text{ dam}$ ($40\text{ m}$):
     $$x^2 + y^2 = 4$$
     This is a concentric circle centered at $(0, 0)$ with radius $r_2 = \sqrt{4} = 2\text{ km}$. (Defined in Level Curves and Contour Maps.)

2. Determine the Vertical Cross-Sectional Trace ($y = 0$): State the Vertical Trace Formulation along Coordinate Planes: Substitute $y = 0$ into the surface equation $z = f(x, y)$:

   $$z = f(x, 0) = x^2 + 0^2 \implies z = x^2$$
   • Geometric Description: In the vertical $xz$-plane, the trace is an upward-opening parabola with its vertex (lowest point) at $(x, z) = (0, 0)$. (Defined in Traces and Cross-Sections of Surfaces.)

3. Slope and Terrain Grade Interpretation:

   • Contour Spacing vs. Gradient: Between $c = 1$ and $c = 4$, elevation increases by $\Delta z = 3\text{ dam} = 30\text{ m}$ over a radial distance of $\Delta r = 2 - 1 = 1\text{ km} = 1000\text{ m}$, yielding an average grade of $\frac{30}{1000} = 3\%$. If we look further out to $c = 9$ ($r = 3\text{ km}$), the elevation increases by another $50\text{ m}$ over $1\text{ km}$ ($5\%$ grade). Because $r = \sqrt{c}$, equal vertical elevation contours produce circular contour rings that grow progressively closer together at larger radii, indicating that the valley walls become increasingly steep further from the origin.

Meaning of the Answer & Real-Life Application:

On a civil engineering contour map, concentric circular level curves indicate a symmetrical bowl or depression. When grading a highway route, transportation engineers examine the vertical trace $z = x^2$ along the proposed road centerline. The instantaneous road grade is given by the derivative of the trace: $\frac{dz}{dx} = 2x$. For example, at $x = 1.5\text{ km}$, $\frac{dz}{dx} = 3\text{ dam/km} = 3\%$, well within typical AASHTO interstate standards ($\le 6\%$). However, at $x = 3.5\text{ km}$, the grade increases to $7\%$, signaling the need for switchbacks, cut-and-fill retaining walls, or tunnel borings to maintain vehicle safety.