Above Earth

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

Correct answer:

h =  4252 km

Step-by-step explanation:

$$\begin{gathered} R=6378\text{ km } \\ {S}_1=4\pi \cdot R^2=4\cdot 3.1416\cdot 6378^2\doteq 511185932.522{\text{km}}^2 \\ S=1\mathrm{/}5\cdot S_1=1\mathrm{/}5\cdot 511185932.522=102237186.505{\text{km}}^2 \\ S=2\pi Rv \\ v=S\mathrm{/}(2\pi \cdot R)=102237186.505\mathrm{/}(2\cdot 3.1416\cdot 6378)={\displaystyle \frac{12756}{5}}=2551.2 \\ y=R-v=6378-2551.2={\displaystyle \frac{19134}{5}}=3826.8 \\ x=\sqrt{R^2-y^2}=\sqrt{6378^2-3826.8^2}={\displaystyle \frac{25512}{5}}=5102.4 \\ k=\text{cotan}\alpha =x\mathrm{/}y=(h+v)\mathrm{/}x \\ x\mathrm{/}y=(h+v)\mathrm{/}x \\ h+v=x^2\mathrm{/}y \\ h=x^2\mathrm{/}y-v=5102.4^2\mathrm{/}3826.8-2551.2=4252\text{ km} \end{gathered}$$

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