A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is $320$, and the sum of the lengths of its $12$ edges is $96$. What is $r$?

1. Let the edges be $a,b,c$. Then $4(a+b+c)=96$, so $a+b+c=24$, and $2(ab+bc+ca)=320$.
2. The space diagonal of the box is the diameter of the sphere: $(2r)^2=a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)=576-320=256$.
3. So $2r=16$ and $r=\boxed{8}$.