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Tree notch (Putnam Exam 1938, rephrased) A notch is cut in
Chapter 7, Problem 61E(choose chapter or problem)
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 \(\theta\). Prove that for a given tree and fixed angle \(\theta\), the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.
Questions & Answers
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 that intersect on a diameter D of the tree. The angle between the two half planes is \(\theta\). Prove that for a given tree and fixed angle \(\theta\), the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.
ANSWER:
Solution Step 1 Consider a cylindrical trunk as shown below. Consider that a notch penetrates to the axis of the cylinder. This is also bounded by two half-planes that intersect on a diameter D. Now, the angle between them is. Consider that the upper part of the notch forms anglewith D and the lower part forms angle with D So, the total angle will be: + = Consider the cut length AB This can be divided as AB = AC + CB ….(1) Here AC and CB are perpendicular to the diameter D. So, you can say AC = D tan CB = D tan Put them into AB = AC + CB AB =