## Task 3 - 160833

In a completely enclosed cuboid glass case, there are exactly $600 \mathrm{~cm}^{3}$ of water. When the case is placed successively on its different outer surfaces on a horizontal plane, the water height in the case is once $2 \mathrm{~cm}$, once $3 \mathrm{~cm}$, and once $4 \mathrm{~cm}$.

Determine the values for the capacity of the case that correspond to these measurements!

Note: The water level is assumed to be part of a horizontal plane, and adhesion is neglected.
If the edge lengths $a, b, c$ (in cm) of the cuboid-shaped interior of the box correspond to the problem statement, then o.B.d.A.

$$
a \cdot b \cdot 2=600 \quad(1) \quad ; \quad a \cdot c \cdot 3=600 \quad(2) \quad ; \quad b \cdot c \cdot 4=600
$$

Dividing (1) by (3), we get $\frac{a}{2 c}=1$ or (4) $a=2 c$. Substituting (4) into (2), we obtain $6 c^{2}=600$, and from this, because $c>0(5) c=10$.

From (5), it follows from (4) that $a=20$ and from (1) or (3) that $b=15$.

Therefore, only $10 \mathrm{~cm}, 15 \mathrm{~cm}, 20 \mathrm{~cm}$ can be the internal dimensions of the box, and thus only the value $3000 \mathrm{~cm}^{3}$ for its capacity can correspond to the problem statement.