Airplane

Aviator sees part of the earth's surface with an area of 200,000 square kilometers. How high he flies?

Correct result:

h =  4.995 km

Solution:

R=6378 km S=200000 km2  S=2πRv v=S/(2π⋅ R)=200000/(2⋅ 3.1416⋅ 6378)≐4.9907 km y=R−v=6378−4.9907≐6373.0093 km x=R2−y2=63782−6373.00932≐252.2639 km k=\cotanα=x/y=(h+v)/x x/y=(h+v)/x h+v=x2/y h=x2/y−v=252.26392/6373.0093−4.9907=4.995 kmR=6378 \ \text{km} \ \\ S=200000 \ \text{km}^2 \ \\ \ \\ S=2 \pi R v \ \\ v=S / (2 \pi \cdot \ R)=200000 / (2 \cdot \ 3.1416 \cdot \ 6378) \doteq 4.9907 \ \text{km} \ \\ y=R - v=6378 - 4.9907 \doteq 6373.0093 \ \text{km} \ \\ x=\sqrt{ R^2-y^2 }=\sqrt{ 6378^2-6373.0093^2 } \doteq 252.2639 \ \text{km} \ \\ k=\cotan \alpha=x/y=(h+v)/x \ \\ x/y=(h+v)/x \ \\ h+v=x^2/y \ \\ h=x^2/y - v=252.2639^2/6373.0093 - 4.9907=4.995 \ \text{km}

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