1. **Problem statement:** Two cars start from the same point and travel along roads forming a 60° angle. Car 1 travels at 50 km/h and Car 2 at 80 km/h. We need to find: i. The distance between the cars after 3 hours. ii. The time when the cars are 315 km apart. 2. **Formula used:** To find the distance between two points moving at an angle, use the Law of Cosines: $$d = \sqrt{a^2 + b^2 - 2ab\cos(\theta)}$$ where $a$ and $b$ are distances traveled by each car, and $\theta$ is the angle between their paths. 3. **Step i: Distance after 3 hours** - Distance traveled by Car 1: $a = 50 \times 3 = 150$ km - Distance traveled by Car 2: $b = 80 \times 3 = 240$ km - Angle: $\theta = 60^\circ$ Calculate: $$d = \sqrt{150^2 + 240^2 - 2 \times 150 \times 240 \times \cos 60^\circ}$$ Since $\cos 60^\circ = 0.5$: $$d = \sqrt{22500 + 57600 - 2 \times 150 \times 240 \times 0.5}$$ $$d = \sqrt{22500 + 57600 - 36000} = \sqrt{44100} = 210$$ km 4. **Step ii: Time when cars are 315 km apart** Let $t$ be the time in hours. Distances: $$a = 50t, \quad b = 80t$$ Distance between cars: $$315 = \sqrt{(50t)^2 + (80t)^2 - 2 \times 50t \times 80t \times \cos 60^\circ}$$ Square both sides: $$315^2 = 2500t^2 + 6400t^2 - 2 \times 50t \times 80t \times 0.5$$ Simplify: $$99225 = 2500t^2 + 6400t^2 - 4000t^2 = 4900t^2$$ Solve for $t^2$: $$t^2 = \frac{99225}{4900} = 20.25$$ $$t = \sqrt{20.25} = 4.5$$ hours **Final answers:** i. After 3 hours, the cars are 210 km apart. ii. The cars will be 315 km apart after 4.5 hours.