Final answer:
The maximum distance between the walkers is found by adding the sum of the radii of both circles to the distance between their centers. Since the radii are 6 and 8 meters, and the centers are 2 meters apart, the maximum distance is 16 meters.
Step-by-step explanation:
The maximum distance between Lucas and Neal while they are walking on their respective circular paths is the distance between the centers of the two circles plus the sum of the radii of the two circles. For Lucas, the center of his circle is at (6,2) and the radius is 6 (since the equation (x−6)^2 +(y−2)^2 = 36 can be rewritten as (x - 6)^2 + (y - 2)^2 = 6^2, where 6 is the radius).
For Neal, the center is at (8,2) and the radius is 8 (since (x−8)^2+(y−2)^2 = 64 can be rewritten as (x - 8)^2 + (y - 2)^2 = 8^2, where 8 is the radius). The distance between the centers of the two circles is the difference in the x-values because the y-values are the same: 8 - 6 = 2 meters.
Now, adding the radii of both circles gives us 6 + 8 = 14 meters. Then, adding this to the distance between the centers, we have 2 + 14 = 16 meters as the maximum distance between Lucas and Neal on their circular paths. The correct answer is d. 16 meters.