1. **State the problem:** Lucas and Neal walk on two circular paths given by the equations: $$ (x + 6)^2 + (y - 5)^2 = 36 $$ and $$ (x - 9)^2 + (y - 5)^2 = 81 $$ We need to find the maximum distance between them at any time. 2. **Identify the centers and radii of the circles:** - First circle center: $(-6, 5)$ with radius $r_1 = \sqrt{36} = 6$ meters. - Second circle center: $(9, 5)$ with radius $r_2 = \sqrt{81} = 9$ meters. 3. **Calculate the distance between the centers:** Since both centers lie on the same horizontal line $y=5$, the distance between centers is: $$ d = |9 - (-6)| = 9 + 6 = 15 \text{ meters} $$ 4. **Find the maximum distance between the walkers:** The maximum distance occurs when both walkers are on the points of their circles farthest apart along the line connecting the centers. This is the sum of the distance between centers and both radii: $$ \text{Max distance} = d + r_1 + r_2 = 15 + 6 + 9 = 30 \text{ meters} $$ **Final answer:** The maximum distance between the walkers is **30 meters**.