If volume of regular tetrahedron of edge length k is V and shortest distance between any pair of opposite edges of same regular tetrahedron is d, then the value of d3V is
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Solution
Length of edge of regular tetrahedron =k
Consider a tetrahedron ABCD
and let A(0,0,0),B(k,0,0),C(k2,√3k2,0)
Projection of point D on plane containing points A,B and C is the centroid of equilateral triangle ABC
Projection of D is (k2,√3k6,0) ∴ Let coordinates of D be (k2,√3k6,h)
|AD|=k ⇒k24+k212+h2=k2 ⇒h2=k2−k23=2k23 ∴h=√2√3k ∴D is (k2,√3k6,√2k√3)
Since, distance between any two skew lines of tetrahedron ABCD=d, ⇒−−→AC⋅−−→CD×−−→AB∣∣∣−−→CD×−−→AB∣∣∣=d ⇒d=(k2^i+√3k2^j)⋅(−2√3k6^j+√2k√3^k)×k^i∣∣
∣∣(−2√36^j+√2k√3^k)×k^i∣∣
∣∣ ⇒d=(k2^i+√3k2^j)⋅(2√3k26^k+√2k2√3^j)k2
⎷(1√3)2+(√2√3)2 =√3k2.√2k2√3k2 ⇒d=1√2k
Area of equilateral ΔABC=√34k2
Volume of tetrahedron, V=13×(ar(ΔABC))×h =13×√34k2×√2√3k=√2k312
∴d3V=(k√2)3(√2k312) =k32√2.12√2k2=124=3
Alternate Solution :
We first draw a cube of side length d and draw a tetrahedron in it.
By Pythagoras theorem, d2+d2=DC2=k2 ⇒d=k√2
Volume of regular tetrahedron with edge length k is V=k36√2 ∴d3V=3