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Cuboid Volume Validation (Posted on 2014-11-30) Difficulty: 3 of 5
The respective minimum distances between an interior diagonal of a rectangular cuboid R and the edges it does not meet are 2√5, 30/√13 and 15/√10.

Determine the volume of the cuboid.

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 1 of 1
Let the cuboid's dimensions be a by b by c and place it on a grid.  Its vertecies will be (0,0,0), (a,0,0), (0,b,0), (0,0,c), (a,b,0), (a,0,c), (0,b,c), (a,b,c).

Define Line D by the point Pd=(0,0,0) and vector Vd=(a,b,c).  This is one of the interior diagonals.
Define Lines A, B and as follows - these will all be skew to line D and to each other:
Line A: Pa = (0,0,c), Va = (a,0,0)
Line B: Pb = (a,0,0), Vb = (0,b,0)
Line C: Pc = (0,b,0), Vc = (0,0,c)

For a pair of skew lines defined by Px,Vx and Py,Vy the orthogonal skew distance is (Px-Py)*(Vx x Vy)/(norm(Vx x Vy).  * is dot product, x is cross product and norm() is the regular norm, or absolute value, of the vector.

Plugging this in for Line D and Line A yields a distance of bc/sqrt(b^2+c^2).  Similarly, Line D and Line B yield ac/sqrt(a^2+c^2) and Line D and Line C yield ab/sqrt(a^2+b^2).

Equate these to each of the given values:
bc/sqrt(b^2+c^2) = 2*sqrt(5)
ac/sqrt(a^2+c^2) = 30/sqrt(13)
ab/sqrt(a^2+b^2) = 15/sqrt(10)

To solve first raise each side of all three equations to the -2 power:
1/b^2 + 1/c^2 = 1/20
1/a^2 + 1/c^2 = 13/900
1/a^2 + 1/b^2 = 2/45

This quickly solves to get a=15, b=5, c=10; the dimensions of the cuboid.

  Posted by Brian Smith on 2016-02-07 23:03:32
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