### 1st Putnam 1938

Problem A4

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. Each half-plane is bounded by a horizontal line passing through the axis of the cylinder. The angle between the two half-planes is θ. Prove that the volume of the notch is minimized (for given tree and θ) by taking the bounding planes at equal angles to the horizontal plane.

Solution

We find the volume of the notch above the horizontal plane. Suppose that the upper bounding half-plane is at an angle φ to the horizontal. We may take the radius of the tree to be 1. A vertical section through the notch at a distance x from its widest extent is a right-angled triangle with base √(1 - x2) and area 1/2 (1 - x2) tan φ. Hence the volume is 2/3 tan φ. So the total volume of the notch is 2/3 (tan φ + tan(θ-φ) ). So we have to find the angle φ which minimises (tan φ + tan(θ - φ). Differentiating, or otherwise, we easily find that the minimum is at φ/2.