1. **Problem statement:** Two planes are at the same altitude. One plane is 100 km away at direction N60°E, and the other is 160 km away at direction S50°E. We need to find the distance between the two planes. 2. **Understanding the directions:** N60°E means 60° east of north, and S50°E means 50° east of south. The angle between these two directions is $$60^\circ + 50^\circ = 110^\circ$$. 3. **Using the Law of Cosines:** To find the distance $d$ between the planes, we use the formula: $$d^2 = a^2 + b^2 - 2ab \cos(C)$$ where $a=100$, $b=160$, and $C=110^\circ$. 4. **Substitute values:** $$d^2 = 100^2 + 160^2 - 2 \times 100 \times 160 \times \cos(110^\circ)$$ 5. **Calculate each term:** $$100^2 = 10000$$ $$160^2 = 25600$$ 6. **Calculate cosine:** $$\cos(110^\circ) = \cos(180^\circ - 70^\circ) = -\cos(70^\circ) \approx -0.3420$$ 7. **Calculate the product:** $$-2 \times 100 \times 160 \times (-0.3420) = 2 \times 100 \times 160 \times 0.3420 = 10944$$ 8. **Sum all terms:** $$d^2 = 10000 + 25600 + 10944 = 46544$$ 9. **Find the distance:** $$d = \sqrt{46544} \approx 215.7$$ km **Final answer:** The planes are approximately 216 km apart.