Water level in a sinking boat is constant? I came across this question in a physics worksheet:

"A small hole is punched into the bottom of a rectangular boat, allowing water to enter the boat. As the boat sinks into the water, which of the following graphs best shows how the rate water flows through the hole varies with time? Assume that the boat remains horizontal as it sinks."

The answer given was a horizontal line, i.e. the rate is constant. The explanation on the answer key was that Archimedes Principle says the difference between the inside and outside water levels is constant, so the flow rate is constant. While I could sort of visualize this, I have never heard of Archimedes Principle used in that way. What's the proof for this (the constant difference in water level)?
 A: Buoyant force equals the density of the fluid times the local acceleration due to gravity times the displaced volume, $F_b = \rho g V$.
Suppose the boat would be floating stationary, thus under no net force, were it not for the hole allowing water through. Then, if the only forces acting on the boat are the buoyant force and gravity, we know $F_b = -F_g = mg$ where m is the mass of the boat.
The mass of the boat is not changing, neither is the density of water, nor the local acceleration due to gravity, so we have
$$mg = \rho g V$$
$$V = \frac{m}\rho$$
And therefore displaced volume is constant. If our boat is a rectangular prism that is sinking straight down, the displacement (the volume of boat that is underwater, minus the volume of boat that is full of water) is
$V = A(h-l)$
where A is the area of the boat, h is the depth relative to surface of the water of the bottom of the boat, and l is the water level in the boat.
Then "the difference between the inside and outside water levels" h-l is given by
$$h-l = \frac{V}A = \frac{m}{A\rho}$$
a constant.

(This only holds if our boat has a flat bottom and vertical walls and is sinking straight down. Other shapes will have constant displacement and constant flow rate, but constant displacement will only correspond to constant difference in water level under such contrived circumstances.)
