Soda cans travel on a conveyor belt at a constant speed. A can is filled with soda upon reaching the belt's midpoint. Each can has a base diameter of 4 inches. The leftmost can is numbered #1, and the rightmost is #15. Express the relationship between the can number (x) and its distance from the filling spout (y) in three different ways.

Let's assume that the distance between each can is equal to the diameter of a can since they are placed next to each other without any gaps. This means that the distance between consecutive cans is 4 inches.

First, we need to establish a point of reference, which is the filling spout. If we consider the filling spout to be at the midpoint of the conveyor belt, we can say that the distance of a can from the filling spout depends on its number.

Let's denote the can number as x, and the distance from the spout as y (in inches).

Now, we establish the relationship for can number 1, the leftmost can. Since it's the first can, there are no cans between it and the midpoint. If can number 1 is currently being filled, it will be exactly at the midpoint, and its distance y from the filling spout will be 0.

For each can to the right of can number 1 (which is getting closer to the spout), the distance from the spout decreases by 4 inches for each increment in the can number. Conversely, for each can to the left of can number 1 (which is moving away from the spout), the distance from the spout increases by 4 inches for each decrement in the can number.

Let's find the relationship for three different situations based on where can #1 is in relation to the filling spout:

1. If can #1 is currently being filled (at the midpoint): For this scenario, as we move to can #2, it will be 4 inches away from the spout; can #3 will be 8 inches, and so on. Hence the relationship would be: y = 4 * (x - 1)

2. If can #1 has just been filled and is now at the point immediately after the midpoint: In this case, can #2 will be at the midpoint, so its distance y from the spout is 0, can #1 is 4 inches away, can #3 will be 4 inches before the spout, and so on. Therefore, we have the following relationship: y = 4 * (2 - x)

3. If can #1 is approaching the spout but has not yet reached the midpoint: This would indicate that each successive can is getting closer to the spout by 4 inches. So the relationship between the can number and its distance from the spout becomes: y = 4 * (x - n) Where n represents the number of the can right at the midpoint. If can #1 is two cans away from being filled, n would be 3.

These equations express the distance of each can from the filling spout as a function of the can number in linear form, with the slope being ±4 depending on the direction of the conveyor belt and the intercept depending on the reference point (can #1's position).