Problem 3

Given tetrahedron

ABCD

, let opposite edges

AB

and

CD

have lengths

a

and

b

. Their skew-line distance is

d

and their angle is

\theta

. A plane parallel to both

AB

and

CD

divides the tetrahedron into two solids. The ratio of the plane's distances from

AB

and

CD

k

. Compute the ratio of the volumes of the two solids.

Step 4 of 5: Add the two pieces

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V_{ABWXYZ}=V_{APWX}+V_{WXPBYZ}=\frac{k^3+3k^2}{(k+1)^3}V=\frac{k^2(k+3)}{(k+1)^3}V.

Detailed analysis

The part of the tetrahedron on the $AB$ side of the original section is $ABWXYZ$ , which is exactly the union of $APWX$ and $WXPBYZ$ . Adding their volumes gives the displayed fraction of $V$ .