smc

Box P has dimensions $l$, $w$, and $h$. Its surface area is $$2lw+2lh+2wh=384,$$ and the sum of all its edges is $$l+w+h=\frac{4l+4w+4h}{4}=\frac{112}{4}=28.$$ The diameter of the sphere is the space diagonal of the prism, which is $$\sqrt{l^2+w^2+h^2}.$$ Notice that $$(l+w+h{)}^2-(2lw+2lh+2wh)=l^2+w^2+h^2=784-384=400,$$ so the diameter is $$\sqrt{l^2+w^2+h^2}=\sqrt{400}=20.$$ The radius is half of the diameter, so $$r=\frac{20}{2}=\boxed{10}.$$