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去年某出版社出版的一本谈数学选择题的解法与训练的书中,在立体几何部份选入了两道有关过圆锥顶点最大截面的选择训练题: 一、圆锥的高为1,底面半径为3~(1/2),过圆锥顶点的截面面积的最大值是: (A)3~(1/2);(B)2;(C)2(3~(1/2));(D)3, 二、己知圆锥的母线长为l,底面半径为R,如果过圆锥顶点的截面面积最大值为l~2/2,那么有: (A)R/l=(2~(1/2))/2; (B)R/l≥(2~(1/2))/2; (C)R/l>(2~(1/2))/2; (D)R/l<(2~(1/2))/2。书中对这两题给出的答案都是(A)。在这里,可能是编者认定“在过圆锥顶点的所有截面中、圆锥的轴截面的面积最大”。并以这一命题为根据作出这样答案的。那么编者们认定的这个命题正确吗?下面我们将对此作一些分析,从而得出相应的结论。
Last year, a book published by a certain publisher on the solution and training of mathematics multiple-choice questions selected two selected training questions on the maximum cross-section of the conical apex in the three-dimensional geometry: 1. The height of the cone is 1, and the bottom The radius is 3~(1/2). The maximum value of the cross-sectional area of the apex of the cone is: (A)3~(1/2); (B)2; (C)2(3~(1/2)) (D)3, Second, the length of the known busbar is l, and the radius of the bottom surface is R. If the maximum cross-sectional area of the cross-cone is l~2/2, then there are: (A) R/l=(2) ~(1/2))/2; (B) R/l≥(2~(1/2))/2; (C)R/l>(2~(1/2))/2; (D R/l<(2~(1/2))/2. The answers to these two questions in the book are (A). Here, the editor may conclude that “the area of the axial section of the cone is the largest in all the cross-sections of the apex of the cone”. And based on this proposition to make such an answer. So are the propositions that the editors identified as correct? Below we will analyze some of them and draw conclusions accordingly.