Volume of Rectangular Box

The areas of the sides of a rectangular box are $24, 32$, and $48\;cm^2$.

What is the volume of this box?
Source: NCTM Mathematics Teacher

SOLUTION

Let $a,b,c$ represent the lengths of the sides.
$ab=24\qquad\left (1\right )$
$ac=32\qquad\left (2\right )$
$bc=48\qquad\left (3\right )$
Eqs. $\left (1\right )$ and $\left (3\right )$ yield
$\frac{ab}{bc}=\frac{24}{48}$
Simplify
$\frac{a}{c}=\frac{1}{2}$
Cross multiply
$c=2a$
Substitute the value of $c$ into Eq. $\left (2\right )$
$a\left (2a\right )=32$
$2a^2=32$
$a^2=16$
$a=4$
Volume of box
$bc=48$
$abc=a\times 48$
$=4\times 48$
$=192$

Answer: $192\;cm^3$