Two mathematically inclined ants are racing to a piece of candy. Ant 1 starts at point M and reaches for a candy at point Q. Ant 2 starts at point A and reaches for a candy at point M. The diagram is not drawn to scale. Ant 1 follows a semicircular path with diameter MN = NO = OP = PQ. Ant 2 follows a square path where AB = BC = CD = AD. MN = AD = X.

1. Express the distance traveled by ant 1 as f(X)

2. Express the distance traveled by ant 2 as g(X)

3. Express the difference between f(X) - g(X) as a function of X

4. Which ant chose the more efficient scenic route?

5. What would have been the most efficient path?

6. If ant 1 and ant 2 reduced the length of their respective path steps (diameter of the semicircle or side of the square to smallest possible step each could take) A) what would be the shape of the path each would take and B) What would be the difference between f(X) and g(X)?

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