Step 1: Name the law.
The flow of a liquid through a thin tube is given by Poiseuille's law. It tells how fast liquid moves under a pushing pressure.
Step 2: Write the formula.
The volume flow rate is \[ Q = \frac{\pi P r^4}{8\eta l} \] where $P$ is the pressure difference, $r$ is the radius, $l$ is the length, and $\eta$ is the viscosity.
Step 3: Pick out radius and length.
The question asks how $Q$ depends on $r$ and $l$. From the formula, $Q$ grows with $r^4$ and falls with $l$. \[ Q \propto \frac{r^4}{l} \]
Step 4: Understand the strong radius effect.
The radius appears to the fourth power. So even a small change in radius changes the flow a lot. A wider tube lets much more liquid through.
Step 5: Rule out the others.
Choices with $r^2$, $r$, or $r^3$ are wrong because they do not match the fourth power that the law demands.
Step 6: State the answer.
So the flow rate is proportional to $r^4/l$. \[ \boxed{Q \propto \frac{r^4}{l}} \]