Airplane

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

Correct answer:

h =  4.9947 km

Step-by-step explanation:

$$\begin{gathered} R=6378\text{ km } \\ S=200000{\text{km}}^2 \\ S=2\pi Rv \\ v=S\mathrm{/}(2\pi \cdot R)=200000\mathrm{/}(2\cdot 3.1416\cdot 6378)\doteq 4.9907\text{ km } \\ y=R-v=6378-4.9907\doteq 6373.0093\text{ km } \\ x=\sqrt{R^2-y^2}=\sqrt{6378^2-6373.0093^2}\doteq 252.2639\text{ km } \\ k=\text{cotan}\alpha =x\mathrm{/}y=(h+v)\mathrm{/}x \\ x\mathrm{/}y=(h+v)\mathrm{/}x \\ h+v=x^2\mathrm{/}y \\ h=x^2\mathrm{/}y-v=252.2639^2\mathrm{/}6373.0093-4.9907=4.9947\text{ km} \end{gathered}$$

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