At a geometric altitude $h_G = 20\text{ km}$ above the Earth's surface (radius $r_0 = 6{,}371\text{ km}$), the corresponding geopotential altitude $h$ (in $\text{km}$, rounded to two decimal places) assuming inverse-square gravitational variation is closest to:

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### 1. Strategy & Theoretical Principles Geopotential altitude $h$ is defined by measuring gravitational potential relative to standard sea-level gravity $g_0$: $$g_0 h = \int_0^{h_G} g(z) \, dz$$ Under Newton's law of universal gravitation, $g(z) = g_0 \left(\frac{r_0}{r_0 + z}\right)^2$. Integrating gives the exact relation: $$h = \frac{r_0 h_G}{r_0 + h_G}$$ where $r_0$ is the mean radius of Earth. ### 2. Analytical Formulation & Calculation Given: - $r_0 = 6{,}371\text{ km}$ - $h_G = 20\text{ km}$ $$h = \frac{6371 \times 20}{6371 + 20} = \frac{127{,}420}{6391} \approx 19.9374\text{ km} \approx 19.94\text{ km}$$ ### 3. Distractor Forensics - For small altitudes $h_G \ll r_0$, $h \approx h_G \left(1 - \frac{h_G}{r_0}\right) = 20 \times (1 - 0.003139) = 19.937\text{ km}$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Geopotential altitude is always slightly *less* than geometric altitude because gravity diminishes with height. - **Standard Reference:** Anderson, J.D., *Introduction to Flight*, Chapter 3 (The Standard Atmosphere).

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