In a laboratory, on a shelf stands a closed glass container in the shape of a cuboid. It contains 2.4 liters of distilled water, but the volume of the container is larger. The water reaches a height of $16 \mathrm{~cm}$. When the cuboid container is placed on another of its sides, the water level will be at a height of $10 \mathrm{~cm}$. If we place it on yet another side, the water will reach a height of $9.6 \mathrm{~cm}$. Determine the dimensions of the container.
Let the dimensions of the container be denoted by $a, b, c$. For each position of the container, we will set up an equation expressing the volume of water:

$$
\begin{aligned}
a \cdot b \cdot 16 & =2400 \\
a \cdot c \cdot 10 & =2400 \\
b \cdot c \cdot 9.6 & =2400
\end{aligned}
$$

We solve the system of three equations with three unknowns:

$$
\begin{aligned}
a \cdot b & =150 \\
a \cdot c & =240, \\
b \cdot c & =250 .
\end{aligned}
$$

From equation (1), we calculate $b$, from equation (2) we calculate $c$, and substitute into equation (3), resulting in the equation

$$
\frac{150}{a} \cdot \frac{240}{a}=250 \text {. }
$$

This equation has a unique solution (in the set of positive numbers) $a=12$. By substituting into the above relationships, we calculate $b, c$.

The dimensions of the container are therefore $12 \text{ cm}, 12.5 \text{ cm}$, and $20 \text{ cm}$.

[setting up the system of equations - 1 pt., calculating one unknown - 2 pts., calculating the second and third unknowns $-1+1$ pt., correct answer - 1 pt.]