imc

Let $X$ be the length of the shortest dimension and $Z$ be the length of the longest dimension. Thus, $XY=24$, $YZ=72$, and $XZ=48$. Divide the first two equations to get $\frac{Z}{X}=3$. Then, multiply by the last equation to get $Z^2=144$, giving $Z=12$. Following, $X=4$ and $Y=6$. The final answer is $4+6+12=22$. $\boxed{22}$