MathLabs

Problem 2

Prove that if one and only one edge of a tetrahedron is greater than 11, then its volume is at most 18\frac18.
Step 2 of 6: Express the volume
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V=16zhAhBsin⁡θV=\frac16z h_Ah_B\sin\theta
Tetrahedron with an opposite edge pair
A translucent 3D tetrahedron illustrating the opposite-edge geometry used in the volume estimate.
Detailed analysis

Let hA,hBh_A,h_B be the perpendicular distances from A and B to line CD, and let θ\theta be the angle between the two face planes through CD. The scalar triple product gives V=16zhAhBsin⁡θV=\frac16z h_Ah_B\sin\theta.