Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units.  Given that the volume of this set is $(m + n \pi)/p$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$.
1. **Calculate the volume of the rectangular parallelepiped (box):**
   The dimensions of the box are \(3 \times 4 \times 5\). Therefore, the volume \(V_{\text{box}}\) is:
   \[
   V_{\text{box}} = 3 \times 4 \times 5 = 60 \text{ cubic units}
   \]

2. **Calculate the volume of the region within one unit of the box:**
   The points external to the box within one unit can be broken up into three types of pieces: rectangular prisms, quarter cylinders, and eighth-spheres.

3. **Calculate the volume of the rectangular prisms:**
   The prisms are formed by extending each face of the box by 1 unit. There are 6 faces, and each face extends to form a rectangular prism of thickness 1 unit. The total volume of these prisms is:
   \[
   2 \times (3 \times 4 + 4 \times 5 + 5 \times 3) = 2 \times (12 + 20 + 15) = 2 \times 47 = 94 \text{ cubic units}
   \]

4. **Calculate the volume of the quarter cylinders:**
   There are 12 quarter cylinders, each with radius 1. Four will have a height of 3, four will have a height of 4, and four will have a height of 5. We can combine them into 3 full cylinders of radius 1 and heights 3, 4, and 5. The volume of these cylinders is:
   \[
   \pi \times 1^2 \times (3 + 4 + 5) = \pi \times 12 = 12\pi \text{ cubic units}
   \]

5. **Calculate the volume of the eighth-spheres:**
   There are 8 eighth-spheres, each with radius 1. Combining them forms a single sphere of radius 1. The volume of this sphere is:
   \[
   \frac{4}{3} \pi \times 1^3 = \frac{4}{3} \pi \text{ cubic units}
   \]

6. **Sum the volumes:**
   Adding the volumes of the box, the prisms, the cylinders, and the sphere, we get:
   \[
   60 + 94 + 12\pi + \frac{4}{3}\pi
   \]

7. **Combine the terms:**
   \[
   60 + 94 = 154
   \]
   \[
   12\pi + \frac{4}{3}\pi = \frac{36\pi}{3} + \frac{4\pi}{3} = \frac{40\pi}{3}
   \]
   Therefore, the total volume is:
   \[
   154 + \frac{40\pi}{3} = \frac{462 + 40\pi}{3}
   \]

8. **Identify \(m\), \(n\), and \(p\):**
   From the expression \(\frac{462 + 40\pi}{3}\), we have \(m = 462\), \(n = 40\), and \(p = 3\).

9. **Calculate \(m + n + p\):**
   \[
   m + n + p = 462 + 40 + 3 = 505
   \]

The final answer is \(\boxed{505}\)