Task 1 - 110831 A vessel (without a drain) with a capacity of 1000 liters was initially filled with exactly 30 liters of water per second at a constant flow rate, and from a later time $t$ onwards, exactly 15 liters of water per second. After exactly $40 \mathrm{~s}$, measured from the beginning, the vessel was full.

Determine what fraction of the vessel's content was filled at time $t$!
The time elapsed from the beginning until time $t$, during which exactly 30 liters per second flowed into the container, is $x$. Then, during the time when exactly 15 liters per second flowed into the container, exactly $(40-x)$ seconds have passed, and it holds that:

$$
30 x+(40-x) \cdot 15=1000
$$

thus $15 x=400$, from which we get $x=\frac{80}{3}$.

During these $\frac{80}{3} \mathrm{~s}$, exactly $\frac{80}{3} \cdot 30 \mathrm{l}$, which is 800 liters of water, flowed into the container. Since $\frac{800}{1000}=\frac{4}{5}$, exactly $\frac{4}{5}$ of the container was filled at time $t$.