jmc

If $l$, $w$, and $h$ represent the dimensions of the rectangular box, we look for the volume $lwh$. We arbitrarily set $lw=24$, $wh=16$, and $lh=6$. Now notice that if we multiply all three equations, we get $l^2{w}^2{h}^2=24\cdot 16\cdot 6=2^3\cdot 3\cdot 2^4\cdot 2\cdot 3=2^8\cdot 3^2$. To get the volume, we take the square root of each side and get $lwh=2^4\cdot 3=\boxed{48}$ cubic inches.