A boat loaded with rocks floats in the middle of a swimming pool. A man in the boat throws the rocks overboard so that they sink to the bottom. How will the water level in the pool change?

Introduction / Context:
This classic hydrostatics question tests your understanding of Archimedes' principle and buoyancy in two different situations: an object in a floating boat and the same object resting on the bottom of the pool. It can be counterintuitive and is often used to check whether students can correctly reason about displaced volume versus actual volume of a denser object like rock.

Given Data / Assumptions:

• A boat loaded with rocks is floating in the middle of a swimming pool.
• The system is initially in equilibrium and the pool water has a certain level.
• The man in the boat throws the rocks into the pool; they sink and rest on the bottom.
• The pool is large enough that changes in level, though conceptually important, are small in practice.

Concept / Approach:
When the rocks are in the boat, the boat plus rocks float according to Archimedes' principle: they displace a volume of water whose weight equals the total weight of the boat plus rocks. The amount of water displaced due to the rocks is therefore equal in weight to the rocks themselves. Since rocks are denser than water, this displaced volume of water is larger than the actual volume of the rocks. When the rocks are thrown overboard and sink, each rock now displaces a volume of water equal to its own volume, not to its weight. Because their volume is smaller than the volume of water that previously had the same weight as the rocks, the total displaced water volume decreases. A smaller displaced volume means the overall water level in the pool must fall.

Step-by-Step Solution:
Step 1: While the rocks are in the floating boat, the boat displaces water of weight equal to the weight of the boat plus rocks.Step 2: Consider only the rocks contribution: they cause a displacement of water whose weight equals the weight of the rocks.Step 3: Because rocks are denser than water, the volume of water that has their weight is greater than the actual volume of the rocks.Step 4: After the rocks are thrown into the pool and sink, each rock displaces a volume of water equal to its own volume.Step 5: This new displaced volume is less than the previously displaced volume due to the rocks while in the boat.Step 6: Therefore, the total volume of displaced water decreases and the water level in the pool decreases accordingly.

Verification / Alternative check:
You can verify this by comparing extreme densities. Imagine objects much denser than water; when floated via a boat, they cause large displaced volumes equal to their weight. But when placed directly in the water, only their smaller actual volume matters. This always reduces the volume of water displaced. Also, textbooks often treat this exact scenario as a standard example, with the correct conclusion that the water level falls after dense objects are removed from a floating vessel and sunk separately.

Why Other Options Are Wrong:
The water level does not increase because the rocks, once sunk, displace less water than before. It does not remain the same because the displacement volume genuinely changes when the rocks move from the boat to the bottom. There is no reason for a non-monotonic change such as first decrease and then increase; the process is straightforward, and the final water level is simply lower than the initial level.

Common Pitfalls:
Many students incorrectly reason that the rocks always displace the same volume of water whether in the boat or at the bottom, which is not true. The key is that in the floating case, the displacement is based on weight, while for sunk objects, it is based on volume. Forgetting the density difference leads to wrong answers. Always think carefully about whether the situation involves floating or sinking when applying Archimedes' principle.

Final Answer:
When the rocks are thrown overboard and sink, the water level in the pool decreases.