HMMT 二月 2003 · 冲刺赛 · 第 31 题

英文解析

31. A cylinder of base radius 1 is cut into two equal parts along a plane passing throughthe center of the cylinder and tangent to the two base circles. Suppose that each piece’ssurface area is m times its volume. Find the greatest lower bound for all possible valuesof m as the height of the cylinder varies.
    Solution: 3
    Let h be the height of the cylinder. Then the volume of each piece is half the volumeof the cylinder, so it is πh . The base of the piece has area π , and the ellipse formed 1
    √2
    h 2
    by the cut has area π · 1 · 1 + because its area is the product of the semiaxes timesπ . The rest of the area of the piece is half the lateral area of the cylinder, so it is πh .4
    Thus, the value of m is
    √
    √
    π + π 1 + h / 4 + πh 22
    2 + 2 h + 4 + hπh/ 2 h=
    √
    2 4 = + 2 + + 1 ,
    h h 2
    a decreasing function of h whose limit as h → ∞ is 3. Therefore the greatest lowerbound of m is 3.
    2 2 2