2015 AMC 10A Problem 21

Tetrahedron ABCD has AB=5, AC=3, BC=4, BD=4, AD=3, and CD=\tfrac{12}5\sqrt2. What is the volume of the tetrahedron?

\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.