Pilot

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

Correct answer:

h =  12.7816 km

Step-by-step explanation:

$$\begin{gathered} R=6378\text{ km } \\ {S}_1=4\pi \cdot R^2=4\cdot 3.1416\cdot 6378^2\doteq 511185932.522{\text{km}}^2 \\ S=0.001\cdot S_1=0.001\cdot 511185932.522\doteq 511185.9325{\text{km}}^2 \\ S=2\pi Rv \\ v=S\mathrm{/}(2\pi \cdot R)=511185.9325\mathrm{/}(2\cdot 3.1416\cdot 6378)={\displaystyle \frac{3189}{250}}=12.756\text{ km } \\ y=R-v=6378-12.756=6365.244\text{ km } \\ x=\sqrt{R^2-y^2}=\sqrt{6378^2-6365.244^2}\doteq 403.1784\text{ km } \\ x\mathrm{/}y=(h+v)\mathrm{/}x \\ h+v=x^2\mathrm{/}y \\ h=x^2\mathrm{/}y-v=403.1784^2\mathrm{/}6365.244-12.756={\displaystyle \frac{6378}{499}}=12.7816\text{ km} \end{gathered}$$

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