Water flows out of a filled cylinder of base area $G$ through a small hole at the bottom. Let $h_0$ denote the initial height of the water at time $t_0 = 0$. Determine the height $h(t)$ of the water level as a function of time $t$. Determine the height $h(t)$ of the water level as a function of time $t$, whereby the rate of change of the volume is proportional to the height $h(t)$.
This problem should be solvable with Torricelli's theorem, I think.
If I want to represent the decrease in level as a function of time, then I only have to rearrange $t=\frac{A}{A_{a}} \sqrt{\frac{2}{g}}(\sqrt{H}-\sqrt{h})$ according to the height $h$, don't I?
$$\frac{A_{a}}{A} \sqrt{\frac{g}{2}} \cdot t=\sqrt{H}-\sqrt{h}$$ $$\sqrt{h}=\sqrt{H}-\frac{A_{a}}{A} \sqrt{\frac{g}{2}} \cdot t$$ $$h=\left(\sqrt{H}-\frac{A_{a}}{A} \sqrt{\frac{g}{2}} \cdot t\right)^{2}$$ $$h(t)=\left(\sqrt{H}-\frac{A_{a}}{A} \sqrt{\frac{g}{2}} \cdot t\right)^{2}$$
Does this consideration fulfil the task, or do I need to look at the rate of change of the volume in more detail, and if so, how?
