1. **State the problem:** We need to find the smallest possible angle $\theta$ between the plane's path and the ground, given the plane reaches a height of 10 km and flies a distance between 57 km and 62 km. 2. **Identify the triangle and variables:** The plane's path is the hypotenuse $h$, the height is the opposite side $o = 10$ km, and the ground distance is the adjacent side $a$. The angle $\theta$ is between the ground and the hypotenuse. 3. **Formula used:** In a right triangle, $\sin(\theta) = \frac{o}{h}$. 4. **Calculate smallest angle:** The smallest angle corresponds to the largest hypotenuse (longest path), which is 62 km. 5. **Calculate $\theta$:** $$\sin(\theta) = \frac{10}{62} \approx 0.16129$$ 6. **Find $\theta$ using inverse sine:** $$\theta = \arcsin(0.16129)$$ 7. **Evaluate $\theta$:** $$\theta \approx 9.3^\circ$$ **Final answer:** The smallest possible angle is approximately **9.3 degrees** to 1 decimal place.