1. **Stating the problem:** Two hikers, Alan and Britney, start at the same campsite. Alan walks 4.5 km north, and Britney walks 6.2 km on a bearing of 65° from Alan's path. We need to find the distance between them after they stop walking. 2. **Formula used:** We use the cosine rule for triangles, which states: $$a^2 = b^2 + c^2 - 2bc \cos A$$ where $a$ is the side opposite angle $A$, and $b$ and $c$ are the other two sides. 3. **Applying the cosine rule:** Here, the distance between Alan and Britney is side $|AB|$, opposite the angle of 65°. $$|AB|^2 = 4.5^2 + 6.2^2 - 2 \times 4.5 \times 6.2 \times \cos 65^\circ$$ 4. **Calculating each term:** $$4.5^2 = 20.25$$ $$6.2^2 = 38.44$$ $$2 \times 4.5 \times 6.2 = 55.8$$ 5. **Substitute values:** $$|AB|^2 = 20.25 + 38.44 - 55.8 \times \cos 65^\circ$$ 6. **Calculate $\cos 65^\circ$:** $$\cos 65^\circ \approx 0.4226$$ 7. **Multiply:** $$55.8 \times 0.4226 \approx 23.58$$ 8. **Final calculation:** $$|AB|^2 = 20.25 + 38.44 - 23.58 = 35.11$$ 9. **Find $|AB|$ by taking the square root:** $$|AB| = \sqrt{35.11} \approx 5.93$$ **Answer:** The distance between Alan and Britney after they stop walking is approximately **5.93 km**.