Question

If volume of a regular tetrahedron of edge length k is V and shortest distance between any pair of opposite edges of same regular tetrahedron is d, then find the value of $\frac{d^3}{V}$

• A. $\frac{\sqrt{6}}{5}$
• B. $\frac{9\sqrt{3}}{4}$
• C. $\frac{6\sqrt{3}}{5}$
• D. $\frac{9\sqrt{6}}{4}$

Answer 1

The correct option is D $\frac{9\sqrt{6}}{4}$

Shortest distance between opposite edges $=\frac{\sqrt{3}}{2}k$
$v=\frac{k^2}{6\sqrt{2}},d=\frac{\sqrt{3}}{2}k$
$d^3=\frac{3\sqrt{3}}{8}{k}^3=\frac{3\sqrt{3}}{8}\ast 6\sqrt{2}v$
$d^3=\frac{9\sqrt{6}}{4}v\Rightarrow \frac{d^3}{v}=\frac{9\sqrt{6}}{4}$