The distance from town A to town B is 5 miles, and town C is 6 miles from town B. Which of the following could be the maximum possible distance (in miles) between towns A and C?

Introduction / Context:
This question tests understanding of the triangle inequality in geometry applied to distances between three towns. If you consider the towns as points on a map, the distances between them must satisfy certain constraints. We are asked about the maximum possible distance between two of the towns when the other two distances are known.

Given Data / Assumptions:

- Distance from town A to town B = 5 miles.
- Distance from town B to town C = 6 miles.
- Distance from town A to town C = AC miles (unknown).
- The three towns can form a straight line or a triangle, as long as distance rules are satisfied.

Concept / Approach:
For any three points A, B, and C, the distance AC must satisfy the triangle inequality: the distance between any two points cannot exceed the sum of the other two distances, and cannot be less than the absolute difference of those distances. That is, |AB - BC| ≤ AC ≤ AB + BC. To find the maximum possible AC, we use the upper bound AB + BC.

Step-by-Step Solution:
AB = 5 miles. BC = 6 miles. Triangle inequality says: |AB - BC| ≤ AC ≤ AB + BC. Compute the absolute difference: |5 - 6| = 1 mile. Compute the sum: 5 + 6 = 11 miles. Therefore 1 ≤ AC ≤ 11. So the maximum possible distance AC is 11 miles.

Verification / Alternative check:
The maximum occurs when the three towns lie on a straight line with B between A and C. In this case, the distance from A to C is simply AB + BC = 5 + 6 = 11 miles. This configuration satisfies all geometric rules and matches the upper bound from the triangle inequality, so 11 miles is indeed achievable and is the maximum.

Why Other Options Are Wrong:
8, 4, and 2 miles: These are all less than 11 miles and so are possible values for AC but not the maximum possible value. The question specifically asks which value could be the maximum distance.

Common Pitfalls:
Sometimes learners confuse the maximum and minimum possible distances or forget to apply the triangle inequality. Another common mistake is to think that AC must be strictly less than AB + BC, but for collinear points it can equal the sum. Always remember that for three points, the range of possible AC values is from the absolute difference to the sum of the two known side lengths.

Final Answer:
The maximum possible distance between towns A and C is 11 miles.