### 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 - x^{2}) and area 1/2 (1 - x^{2}) 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.

1st Putnam 1938

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002