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 $\frac{bc}{ad}?$

We are given a right rectangular prism $ B $ with edge lengths 1, 3, and 4. We need to determine the volume of the set $ S(r) $ of points within a distance $ r $ from the prism $ B $, and express this volume in the form $ ar^3 + br^2 + cr + d $, and then compute $ \frac{bc}{ad} $.

### Step 1: Volume of the Rectangular Prism
The volume of the original rectangular prism $ B $ is the product of its edge lengths:
\[
V_B = 1 \times 3 \times 4 = 12
\]
This corresponds to the constant term $ d $ in the polynomial expression for the volume of $ S(r) $.

### Step 2: Contribution of the Faces
The prism has 6 faces. Each face contributes to the volume when the distance $ r $ is considered. The area of each face is:
- 2 faces with area $ 1 \times 3 = 3 $
- 2 faces with area $ 1 \times 4 = 4 $
- 2 faces with area $ 3 \times 4 = 12 $

The total surface area $ A $ of the prism is:
\[
A = 2(3 + 4 + 12) = 38
\]
When extending each face outward by distance $ r $, it contributes a volume $ r $ times the surface area. Therefore, the total contribution from all faces is:
\[
\text{Volume contribution from faces} = A \times r = 38r
\]
This corresponds to the linear term $ c $ in the polynomial.

### Step 3: Contribution of the Edges
The prism has 12 edges. Each edge contributes to the volume by extending radially outward to form a quarter-cylinder along its length. The length of each edge is:
- 4 edges of length 1,
- 4 edges of length 3,
- 4 edges of length 4.

The total length $ L $ of all edges is:
\[
L = 4(1 + 3 + 4) = 32
\]
The volume of a quarter-cylinder of radius $ r $ and height $ h $ is $ \frac{\pi r^2 h}{4} $, so the total contribution from all edges is:
\[
\text{Volume contribution from edges} = L \times \frac{\pi r^2}{4} = \frac{32\pi r^2}{4} = 8\pi r^2
\]
This corresponds to the quadratic term $ b $ in the polynomial.

### Step 4: Contribution of the Vertices
The prism has 8 vertices. Around each vertex, the volume contribution is a quarter of a sphere with radius $ r $. The volume of a quarter-sphere is $ \frac{\pi r^3}{8} $, so the total contribution from all vertices is:
\[
\text{Volume contribution from vertices} = 8 \times \frac{\pi r^3}{8} = \pi r^3
\]
This corresponds to the cubic term $ a $ in the polynomial.

### Step 5: Coefficients and the Expression for Volume
Summarizing, the volume $ V(r) $ of $ S(r) $ can be expressed as:
\[
V(r) = \pi r^3 + 8\pi r^2 + 38r + 12
\]
Thus, the coefficients are $ a = \pi $, $ b = 8\pi $, $ c = 38 $, and $ d = 12 $.

### Step 6: Computing $ \frac{bc}{ad} $
Finally, we compute:
\[
\frac{bc}{ad} = \frac{(8\pi) \times 38}{\pi \times 12} = \frac{304\pi}{12\pi} = \frac{304}{12} = \frac{76}{3}
\]
Thus, the value of $ \frac{bc}{ad} $ is:
\[
\boxed{\frac{76}{3}}
\]

Calculate the volume of the set of points within a given distance from a rectangular prism with edge lengths 1, 3, and 4. Understand how different geometric shapes contribute to the total volume using mathematical formulas. Suitable for advanced high school students and beyond.

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