The areas of three adjacent faces of a cuboidal box are $96\, cm^2, 48\, cm^2$ and $72\, cm^2$, respectively. The volume of the box is

$576\, cm^3$

The correct answer is Option (2) → $576\, cm^3$

1. Identify the given information

Let the dimensions of the cuboid be length ($l$), width ($w$), and height ($h$). The areas of the three adjacent faces are:

1. $l \times w = 96 \text{ cm}^2$
2. $w \times h = 48 \text{ cm}^2$
3. $h \times l = 72 \text{ cm}^2$

2. Formulate the relationship

The volume ($V$) of a cuboid is given by:

$V = l \times w \times h$

If we multiply the areas of the three adjacent faces together, we get:

$(l \times w) \times (w \times h) \times (h \times l) = l^2 \times w^2 \times h^2 = (l \times w \times h)^2$

This means:

$\text{Product of areas} = V^2$

3. Calculate the Volume

Substitute the given areas into the formula:

$V^2 = 96 \times 48 \times 72$

$V^2 = 331,776$

Now, take the square root to find the volume:

$V = \sqrt{331,776}$

$V = 576 \text{ cm}^3$

Final Answer:

The volume of the box is $576 \text{ cm}^3$.