Hillside Foundation Slope Stability (Civil & Construction Engineering)

A geotechnical and civil engineering firm assesses hillside terrain stability prior to constructing a reinforced concrete retaining wall and foundation slab for a commercial development in Henderson, Nevada.

A drone-based LiDAR survey generates a continuous digital elevation model $E(x, y)$ (in meters, $\text{m}$) across the building pad:

where $x$ represents the distance eastward (in meters) and $y$ represents the distance northward (in meters) relative to an anchored geodetic survey monument.

1. State how cross-sectional vertical cuts $y = y_0$ and $x = x_0$ define directional slopes on the terrain surface.
2. Determine the algebraic expressions for the directional slopes $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$.
3. Evaluate the slopes $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$ at the proposed foundation center location $(x, y) = (4, 10)\text{ m}$.
4. State whether the ground slopes uphill or downhill when walking due east ($+x$) and due north ($+y$) from the center location, and state the angle of inclination along each cardinal direction in degrees ($\theta = \arctan(m)$).