Pilot

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

Result

h =  12.782 km

Solution:

R=6378 km S1=4π⋅ R2=4⋅ 3.1416⋅ 63782≐511185932.5225 km2 S=0.001⋅ S1=0.001⋅ 511185932.5225≐511185.9325 km2  S=2πRv v=S/(2π⋅ R)=511185.9325/(2⋅ 3.1416⋅ 6378)=3189250=12.756 km y=R−v=6378−12.756=6365.244 km x=R2−y2=63782−6365.2442≐403.1784 km  x/y=(h+v)/x h+v=x2/y h=x2/y−v=403.17842/6365.244−12.756=6378499≐12.7816=12.782  km R = 6378 \ km \ \\ S_{ 1 } = 4 \pi \cdot \ R^2 = 4 \cdot \ 3.1416 \cdot \ 6378^2 \doteq 511185932.5225 \ km^2 \ \\ S = 0.001 \cdot \ S_{ 1 } = 0.001 \cdot \ 511185932.5225 \doteq 511185.9325 \ 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 \ km \ \\ y = R - v = 6378 - 12.756 = 6365.244 \ km \ \\ x = \sqrt{ R^2-y^2 } = \sqrt{ 6378^2-6365.244^2 } \doteq 403.1784 \ 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 } \doteq 12.7816 = 12.782 \ \text{ km }

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