Adrian Distance 80C6Ba

1. **State the problem:** Adrian flies $357$ km due south from Darwin, then flies on a bearing of $N29^\circ W$ until he is due west of Darwin. We need to find how far Adrian is from Darwin now. 2. **Understand the scenario:** - First leg: $357$ km south. - Second leg: flies on bearing $N29^\circ W$ until directly west of Darwin. 3. **Set up coordinate system:** - Place Darwin at origin $(0,0)$. - After first leg, Adrian is at point $A = (0, -357)$. 4. **Second leg direction:** - Bearing $N29^\circ W$ means $29^\circ$ west of north. - The direction vector has components: $$x = -\sin 29^\circ, \quad y = \cos 29^\circ$$ 5. **Let the distance flown on second leg be $d$ km.** - Coordinates after second leg: $$x = 0 + d(-\sin 29^\circ) = -d \sin 29^\circ$$ $$y = -357 + d \cos 29^\circ$$ 6. **Condition: Adrian is due west of Darwin after second leg.** - Being due west means $y=0$. 7. **Solve for $d$:** $$0 = -357 + d \cos 29^\circ$$ $$d \cos 29^\circ = 357$$ $$d = \frac{357}{\cos 29^\circ}$$ 8. **Calculate $d$: (using $\cos 29^\circ \approx 0.8746$)** $$d \approx \frac{357}{0.8746} \approx 408.2 \text{ km}$$ 9. **Find Adrian's final position coordinates:** $$x = -d \sin 29^\circ \approx -408.2 \times 0.4848 = -197.9$$ $$y = 0$$ 10. **Distance from Darwin now:** - Since Adrian is at $(-197.9, 0)$, distance is $197.9$ km. 11. **Final answer:** - To the nearest km, Adrian is $198$ km from Darwin now.