The spacecraft

The spacecraft spotted a radar device at an altitude angle alpha = 34 degrees 37 minutes and had a distance of u = 615 km 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 6378 km.

Final Answer:

x =  368.3833 km

Step-by-step explanation:

$$\begin{gathered} \alpha =34+\frac{37}{60}=\frac{34\cdot 60}{60}+\frac{37}{60}=\frac{2040}{60}+\frac{37}{60}=\frac{2040+37}{60}=\frac{2077}{60}\doteq 34.6167{}^{\circ } \\ u=615\text{km} \\ R=6378\text{km} \\ \beta =90+\alpha =90+\frac{2077}{60}=\frac{90\cdot 60}{60}+\frac{2077}{60}=\frac{5400}{60}+\frac{2077}{60}=\frac{5400+2077}{60}=\frac{7477}{60}\doteq 124.6167{}^{\circ } \\ (x+R{)}^2=u^2+R^2-2\cdot u\cdot R\cdot \cos \alpha \\ x=\sqrt{u^2+R^2-2\cdot u\cdot R\cdot \cos \beta }-R=\sqrt{u^2+R^2-2\cdot u\cdot R\cdot \cos 124.61666666667°}-R=\sqrt{615^2+6378^2-2\cdot 615\cdot 6378\cdot \cos 124.61666666667°}-6378=\sqrt{615^2+6378^2-2\cdot 615\cdot 6378\cdot (-0.568083)}-6378=368.383=368.3833\text{km} \end{gathered}$$

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