Let \mathcal{R} be the region consisting of the set of points in the coordinate plane that satisfy both |8-x|+y \leq 10 and 3 y-x \geq 15. When \mathcal{R} is revolved around the line whose equation is 3 y-x=15, the volume of the resulting solid is \frac{m \pi}{n \sqrt{p}}, where m, n, and p are positive integers, m and n are relatively prime, and p is not divisible by the square of any prime. Find m+n+p.