2020 AMC 10B Problem 20

Let B be a right rectangular prism (box) with edges lengths 1, 3, and 4, together with its interior. For real r\geq0, 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 ar^{3} + br^{2} + cr +d, where a, b, c, and d are positive real numbers. What is \frac{bc}{ad}?

\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.