Solution: Math Puzzle: Float and Sink

Solution: Float and Sink Math Puzzle

You can think of this as a two-step process. First remove the stone from the boat. This will cause the boat to rise, since it now weighs less and is displacing less water. Because the boat displaces less water, the pond level goes down as the boat rises.

Now when you drop the stone into the pond, the water level rises again, but by how much? It rises less than the level dropped when the stone was removed from the boat. The reason is that the boat with the stone in it displaces more water than the boat without the stone plus the submerged stone.

Let’s put some numbers to this. Let’s say that the boat with just the passenger has a mass of 100 kg. According the Archimedes’ principle, it will displace 100 kg of water, which equals a volume of 100 L.

With a 19 kg stone on board, the boat now displaces 119 kg of water with a volume of 119 L. When we remove the stone from the boat, it again displaces 100 L of water. When we put the stone in the pond, it displaces some water, but how much?

Since the stone is denser than water, it sinks. When an object sinks, it displaces a volume of water equal to its size. The density of granite varies, but it is about 2.75 g/cm³. Water’s density is 1.0 g/cm³. We can calculate the volume that a 19 kg piece of granite would occupy:

V = 19,000 / 2.75
= 6909 cm³
= 6.909 L

With the stone overboard, the total amount of water displaced is 100 L + 6.909 L = 106.909 L, as compared with the 119 L displaced by the boat with the stone in it. Since less water is displaced, the pond level is lower.