Geometry #3

by sjaelee, Aug 22, 2011, 1:43 PM

Quote:
The shortest distances between an interior diagonal of a rectangular parallelepiped (box) P and the edges it does not meet are
$2\sqrt{5}, 30/\sqrt{13}$ and $15/\sqrt{10}$.
Determine the volume of P.

Source: AIME

After drawing the diagram, we find the distances are:

The line to the plane parrallel to the diagnol, meaning the altidue of a right triangle with two sides of the box. (poor explanation, I can't draw a diagram)

Thus we have, WLOG,

$\frac{ab}{\sqrt{a^2+b^2}}=2\sqrt{5}$

$\frac{ac}{\sqrt{a^2+c^2}}=\frac{30}{\sqrt{13}}$

$\frac{bc}{\sqrt{b^2+c^2}}=\frac{15}{\sqrt{10}}$

Note that $1^2+2^2=5,2^2+3^2=13,1^2+3^2=10$, so we scale by $5$ and find

$5,10,15$ work, so

$5*10*15=750$.
This post has been edited 1 time. Last edited by djmathman, Aug 27, 2011, 5:09 PM

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