How high is the airplane's pilot to see 0.001 of Earth's surface?

Correct result:

h =  12.782 km


R=6378 km S1=4π R2=4 3.1416 63782511185932.522 km2 S=0.001 S1=0.001 511185932.522511185.9325 km2  S=2πRv v=S/(2π R)=511185.9325/(2 3.1416 6378)=3189250=12.756 km y=Rv=637812.756=6365.244 km x=R2y2=637826365.2442403.1784 km  x/y=(h+v)/x h+v=x2/y h=x2/yv=403.17842/6365.24412.756=6378499=12.782 kmR=6378 \ \text{km} \ \\ S_{1}=4 \pi \cdot \ R^2=4 \cdot \ 3.1416 \cdot \ 6378^2 \doteq 511185932.522 \ \text{km}^2 \ \\ S=0.001 \cdot \ S_{1}=0.001 \cdot \ 511185932.522 \doteq 511185.9325 \ \text{km}^2 \ \\ \ \\ S=2 \pi R v \ \\ v=S / (2 \pi \cdot \ R)=511185.9325 / (2 \cdot \ 3.1416 \cdot \ 6378)=\dfrac{ 3189 }{ 250 }=12.756 \ \text{km} \ \\ y=R - v=6378 - 12.756=6365.244 \ \text{km} \ \\ x=\sqrt{ R^2-y^2 }=\sqrt{ 6378^2-6365.244^2 } \doteq 403.1784 \ \text{km} \ \\ \ \\ x/y=(h+v)/x \ \\ h+v=x^2/y \ \\ h=x^2/y - v=403.1784^2/6365.244 - 12.756=\dfrac{ 6378 }{ 499 }=12.782 \ \text{km}

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