A triangle $ABC$ with $AB=BC=5$ and $CA=6$ is placed in space with $A$ on the positive $x$-axis, $B$ on the positive $y$-axis, and $C$ on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$?

1. Let $A=(x,0,0)$, $B=(0,y,0)$, $C=(0,0,z)$. Then $x^2+y^2=AB^2=25$, $y^2+z^2=BC^2=25$, $z^2+x^2=CA^2=36$.
2. Adding gives $x^2+y^2+z^2=43$, so $z^2=18$, $x^2=18$, $y^2=7$.
3. $OA,OB,OC$ are mutually perpendicular, so $V=\frac16xyz=\frac16\sqrt{18\cdot7\cdot18}=\frac{18\sqrt7}6=\boxed{3\sqrt7}$.