  In this paper we give an affirmative answer to the following question posed
by Daryl Cooper: If one lengthens the sides of a tetrahedron by one unit, is
the result still a tetrahedron and (if so) does the volume increase? Our proof
involves a (presumably) new and sharp inequality involving the Cayley-Menger
determinant and one of its directional derivatives. We give a rigorous
computer-assisted proof of the inequality. We also sketch an argument which
derives the existence portion of the result, in all dimensions, from an old
theorem of Von Neumann. Finally, we prove a number of additional results
concerning the effect on volume of selectively lengthening some of the sides of
a tetrahedron.
