Let $B$ be a right rectangular prism (box) with edge lengths $1$, $2$, and $2$, together with its interior. For real $r\ge 0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $a{r}^3+b{r}^2+cr+d$, where $a,b,c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}$?