Question

Why would a boat with a hole in the bottom would eventually sink?

Answer 1

Water would enter the boat through hole, replacing the air in the boat. After sometime the boat would no longer be less than water and it would result in sinking. Because we know that for floating conditions the density of the object must be lesser than the liquid.

Complete step by step answer:
The air pressure inside the boat is the same as atmospheric pressure. The pressure in the water at the depth of the whole is greater than that due to the weight of the water.
The pressure difference between the pressure at bottom of the boat and atmosphere is what forces water to enter into the boat.
We know that pressure is defined as force per unit area,
i.e., pressure is inversely proportional to the area $P \propto \dfrac{1}{{area}}$, thus smaller the hole greater will be pressure of water flow and the larger the hole less pressure resulting in faster water entry into the boat.
The water from around the boat enters through hole, if the material of the boat is heavier than water, or the materials in the boat will make it heavier than water then by Archimedes principle we know that object immersed in water will displace equal amount of water as of its weight thus the buoyancy force gradually decreases and the boat will sink.

Note: The air that is in the boat is lighter than the water that takes its place, which makes the boat ride lower until it sinks. The water entered into the boat because water is denser than air and will displace it.