子夜之狼
幼苗
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设该圆锥底面直径为D,高为H
圆柱体底面直径为d,高为h
d:D = (H-h):H
h=(D-d)H/D
圆柱体体积:V=πd^2h / 4 = πd^2 * [(D-d)H/D] /4 = πH/(4D) * (-d^3+Dd^2)
V'=πH/(4D) * (-3d^2+2Dd)
d<2D/3时,V'>0,V单调增;
d>2D/3时,V'<0,V单调减.
∴d=2D/3时,V最大
圆柱体最大体积:Vmax = (πH/4D)【-(2D/3)^3+D(2D/3)^2】= πD^2H/27
圆锥体体积:πR^2H/3 = πD^2H/12
(πD^2H/27) :(πD^2H/12) = 12 / 27 =4 / 9,得证
1年前
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