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 .

do you mean the edges and once you fix edges>=4 you cannot vary the volume — Archis Welankar, Apr 07 '17 at 16:48
Question P http://dc.allenbpms.in/testpaper/solution/d9482-49-636271921434046594-049s.gif @DougM — Koolman, Apr 07 '17 at 16:49
@ArchisWelankar I was fine , but I could hve achieved more . — Koolman, Apr 07 '17 at 16:54

score 2 · Accepted Answer · answered Apr 07 '17 at 16:56

2

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.

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

answered Apr 07 '17 at 16:56

Doug M

I could not understand how do yu get the heig)t – Koolman Apr 07 '17 at 17:01
Visualize the two equilateral connected by a hinge. The 6th edge will connect the free vertexes. Place one triangle on the table, and move the other triangle on its hinge, paying attention to the position of the vertex. At some point it reaches its maximum height, and that height is the distance across the face of that triangle. – Doug M Apr 07 '17 at 17:05

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