Final answer:
The flux of water out of the bucket is negative when it is being held normally to hold water and zero when the bottom of the bucket is cut out.
Step-by-step explanation:
The flux of water out of the bucket when it is being held normally to hold water is negative. This is because the rain is entering the bucket, which means the water is flowing into the bucket and not out of it.
When you cut out the bottom of the bucket and hold it in the rain, the flux of water out of the bucket would be zero. This is because there is no surface for the water to flow out of. Without a bottom, the rainwater will simply pass through the bucket without any accumulation inside.