Tree notch (Putnam Exam 1938, rephrased) 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 diam? eter? of the tree. The angle between the two half pl? anes 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 ? .
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 = D tan + D tan = D (tan + tan ) …(2) So, the area of the triangle is 1 area = 2 (base)(height) 1 area = 2 AB.D 1 = 2(D (tan + tan )).D 2 area = D2 (tan + tan ) If the density of the stem is, then the volume is V = (density)(area) 2 V = ( D (tan + tan ) 2 2 D V = 2 (tan + tan ) ….(3) Now, you know that total angle is + = So, rearrange this in terms of = D2 Put this into the equation V = 2 (tan + tan ) 2 V = D (tan + tan ( )) 2 Now V is in terms of single variable Find the derivative with respect to dV d D2 d = d ( 2 (tan + tan ( )) 2 = (D (sec + sec ( ) d ( )) 2 d D 2 2 2 = ( 2 (sec + sec ( )(0 1)) 2 V = (D (sec sec ( )) 2 Now, in order to minimize the volume equate V to 0 D2 2 2 ( 2 (sec sec ( )) = 0 2 2 sec =sec ( ) 2 sec= ± ec ( ) sec=± sec ( ) Take the positive root sec = sec ( ) Compare the angles on both the sides = 2 = = 2 Put this into the equation = = 2 = 2 So, at the minimum volume of notch = , = 2 2 So, this is proved that for a fixed angle, the volume of the notch is minimized by taking the bounding planes at equal angles to horizontal plane that also passes through D.