Variable volume of tetrahedron

Question

Let the 5 sides of terahedron be 1 . And the sixth side is x .

Now how can we comment that how the volume of tetrahedron with varying x .

When it gets maxima .

Answer 1

2 of the faces of the tetrahedron are equilateral triangles.

Call one of these the base.

$V = \frac 13 b h\\ b = \frac {\sqrt 3}{4}$

To maximise V we must maximize h.

The highest the the remaining vertex can be above the base is if the edge with these two equilateral triangles meets at a right angle.

$V_{max} = \frac 13 \frac {\sqrt 3}{4} \frac {\sqrt 3}{2}\\ V_{max} = \frac 18$