Supersonic Nose Cone Aerodynamic Pressure (Aerospace & Mechanical Engineering)

An aerospace aerodynamicist evaluates the supersonic airflow across the nose cone of a high-speed atmospheric flight test vehicle. In wind tunnel validation studies, the upper curved aerodynamic profile of the fuselage nose cone is modeled by the paraboloid of revolution:

where coordinates $x, y, z$ are measured in centimeters ($\text{cm}$).

During supersonic Mach 2 flight, a pitot-static pressure sensor probe is installed at the surface location $(x_0, y_0) = (10, 5)\text{ cm}$.

1. Compute the exact height $z_0 = f(10, 5)$ of the fuselage at the sensor installation point.
2. Determine the general analytical expressions for the partial derivatives $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.
3. Evaluate the partial derivatives at $(10, 5)$ and state the equation of the tangent plane to the fuselage surface at this location in standard form.
4. Extract the outward-pointing unit normal vector $\hat{n}$ to the fuselage surface at $(10, 5)$, and explain why aerodynamicists use $\hat{n}$ to compute the local static pressure force vector $\vec{F}_p = -p \hat{n} \, dA$.