Woodworking Again

Geometry Level 5

A person finds a piece of wood in a shape of irregular tetrahedron with edges of length in cm: \( AB=7, AC=5, BC=6, AD=8, BD=10, CD=9 \).

Because the surface of the wooden block is damaged, the person decides to take: \(1\) cm off face \(ABC\), \(0.75\) cm off face \(ACD\), \(0.5\) cm off face \(BCD\) and \(0.25\) cm off face \(ABD\). After doing so he notices further damage near the vertices of the tetrahedron so he decides to make a sphere shape. What is the radius \(r\) of the largest sphere he can make of the remaining wooden block?

Submit your answer as \(\lfloor 10000 r \rfloor\).

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