imc

Split $S(r)$ into 4 regions: 1. The rectangular prism itself 2. The extensions of the faces of $B$ 3. The quarter cylinders at each edge of $B$ 4. The one-eighth spheres at each corner of $B$ Region 1: The volume of $B$ is $1\cdot 3\cdot 4=12$, so $d=12$. Region 2: This volume is equal to the surface area of $B$ times $r$ (these "extensions" are just more boxes). The volume is then $\text{SA}\cdot r=2(1\cdot 3+1\cdot 4+3\cdot 4)r=38r$ to get $c=38$. Region 3: We see that there are 12 quarter-cylinders, 4 of each type. We have 4 quarter-cylinders of height 1, 4 quarter-cylinders of height 3, 4 quarter-cylinders of height 4. Since 4 quarter-cylinders make a full cylinder, the total volume is $4\cdot \frac{1\pi r^2}{4}+4\cdot \frac{3\pi r^2}{4}+4\cdot \frac{4\pi r^2}{4}=8\pi r^2$. Therefore, $b=8\pi$. Region 4: There is an eighth-sphere of radius $r$ at each corner of $B$. Since there are 8 corners, these eighth-spheres add up to 1 full sphere of radius $r$. The volume of this sphere is then $\frac{4}{3}\pi \cdot r^3$, so $a=\frac{4\pi }{3}$. Using these values, $\frac{bc}{ad}=\frac{(8\pi )(38)}{(4\pi /3)(12)}=\boxed{19}$.