7. A container in the shape of a rectangular box is partially filled with $120 \mathrm{~m}^{3}$ of water. The depth of the water is either $2 \mathrm{~m}, 3 \mathrm{~m}$ or $5 \mathrm{~m}$, depending on which side of the box is on the ground.
What is the volume of the container in $\mathrm{m}^{3}$ ?
SolUtion
240

Let the dimensions of the box in metres be $x, y, z$, with $x<y<z$. We know $5 x y=3 x z=$ $2 y z=120$. Therefore, $5 x y \times 3 x z \times 2 y z=120^{3}$, giving equation $30 x^{2} y^{2} z^{2}=120^{3}$.
We deduce that $x^{2} y^{2} z^{2}=\frac{120 \times 120 \times 120}{30}=4 \times 120 \times 120$.
Therefore, the volume, in $\mathrm{m}^{3}$, of the container is $x y z=2 \times 120=240$.