The spacecraft

The spacecraft spotted a radar device at altitude angle alpha = 34 degrees 37 minutes and had a distance of u = 615km from Earth's observation point.
Calculate the distance d of the spacecraft from Earth at the moment of observation. Earth is considered a sphere with a radius of 6378km.

Result

x =  368.383 km

Solution:

$α=34 +\dfrac{ 37 }{ 60 }=\dfrac{ 2077 }{ 60 } \doteq 34.6167 \ ^\circ \ \\ \ \\ u=615 \ \text{km} \ \\ R=6378 \ \text{km} \ \\ β=90 + α=90 + 34.6167 \doteq \dfrac{ 7477 }{ 60 } \doteq 124.6167 \ ^\circ \ \\ \ \\ (x+R)^2=u^2 + R^2 - 2 \cdot \ u \cdot \ R \cdot \ \cos α \ \\ \ \\ x=\sqrt{ u^2 + R^2 - 2 \cdot \ u \cdot \ R \cdot \ \cos(β) } - R=\sqrt{ 615^2 + 6378^2 - 2 \cdot \ 615 \cdot \ 6378 \cdot \ \cos(124.6167^\circ ) } - 6378 \doteq 368.3833 \doteq 368.383 \ \text{km}$

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