The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this octahedron?

$\textbf{(A) } \dfrac{75}{12} \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 10\sqrt2 \qquad\textbf{(E) } 15$
The octahedron is just two congruent pyramids glued together by their base. The base of one pyramid is a rhombus with diagonals $4$ and $5$, for an area $A=10$. The height $h$, of one pyramid, is $\frac{3}{2}$, so the volume of one pyramid is $\frac{Ah}{3}=5$. Thus, the octahedron has volume $2\cdot5=\boxed{\textbf{(B) }10}$