An azimuth

The patrol had started at a designated marching angle (an azimuth) of 13°. After 9 km, the azimuth's angle changed to 62°. The patrol went 10 km in this direction. Find the distance from where the patrol started.

Final Answer:

d =  17.2942 km

Step-by-step explanation:

$$\begin{gathered} \alpha =13{}^{\circ } \\ a=9\text{km} \\ \beta =62{}^{\circ } \\ b=10\text{km} \\ {x}_1=a\cdot \sin \alpha =a\cdot \sin 13°=9\cdot \sin 13°=9\cdot 0.224951=2.02456\text{km} \\ {y}_1=a\cdot \cos \alpha =a\cdot \cos 13°=9\cdot \cos 13°=9\cdot 0.97437=8.76933\text{km} \\ {x}_2=b\cdot \sin \beta =b\cdot \sin 62°=10\cdot \sin 62°=10\cdot 0.882948=8.82948\text{km} \\ {y}_2=b\cdot \cos \beta =b\cdot \cos 62°=10\cdot \cos 62°=10\cdot 0.469472=4.69472\text{km} \\ x=x_1+x_2=2.0246+8.8295\doteq 10.854\text{km} \\ y=y_1+y_2=8.7693+4.6947\doteq 13.464\text{km} \\ d=\sqrt{x^2+y^2}=\sqrt{10.854^2+13.464^2}=17.2942\text{km} \end{gathered}$$

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