A notch is cut in a cylindrical vertical tree trunk The notch penetrates to the axis of the cylinder and is bounded by two half planes that intersect on a diameter D of the tree The angle between the two half planes is Prove that for a given tree and fixed angle , the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D DI

Question: A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes

A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is θ. Prove that for a given tree and fixed angle θ, the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.

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