Above Earth

To what height must a boy be raised above the earth in order to see one-fifth of its surface.

Result

h =  4252 km

Solution:

R=6378 km S1=4π⋅ R2=4⋅ 3.1416⋅ 63782≐511185932.5225 km2 S=1/5⋅ S1=1/5⋅ 511185932.5225≐102237186.5045 km2 S=2πRv v=S/(2π⋅ R)=102237186.5045/(2⋅ 3.1416⋅ 6378)=127565=2551.2 y=R−v=6378−2551.2=191345=3826.8 x=R2−y2=63782−3826.82=255125=5102.4 k=\cotanα=x/y=(h+v)/x x/y=(h+v)/x h+v=x2/y h=x2/y−v=5102.42/3826.8−2551.2=4252 kmR=6378 \ \text{km} \ \\ S_{1}=4 \pi \cdot \ R^2=4 \cdot \ 3.1416 \cdot \ 6378^2 \doteq 511185932.5225 \ \text{km}^2 \ \\ S=1/5 \cdot \ S_{1}=1/5 \cdot \ 511185932.5225 \doteq 102237186.5045 \ \text{km}^2 \ \\ S=2 \pi R v \ \\ v=S / (2 \pi \cdot \ R)=102237186.5045 / (2 \cdot \ 3.1416 \cdot \ 6378)=\dfrac{ 12756 }{ 5 }=2551.2 \ \\ y=R - v=6378 - 2551.2=\dfrac{ 19134 }{ 5 }=3826.8 \ \\ x=\sqrt{ R^2-y^2 }=\sqrt{ 6378^2-3826.8^2 }=\dfrac{ 25512 }{ 5 }=5102.4 \ \\ k=\cotan \alpha=x/y=(h+v)/x \ \\ x/y=(h+v)/x \ \\ h+v=x^2/y \ \\ h=x^2/y - v=5102.4^2/3826.8 - 2551.2=4252 \ \text{km}

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