推 newperson :懂了~感謝^^ 05/04 08:55
※ 引述《newperson (123456)》之銘言：
: 一個球的內接圓錐的最大體積與這個球的體積比為?
: 答案是 8:27
: 請指教 謝謝
obviously, the right cone is bigger than slant cone under
the same base disk.
assume the radius of disk u, the radius of shpere 1,
and the height of the cone s.
then the volume of the cone V is
V=πu^2 s/3
however, (s-1)^2+u^2=1
Let s=1+cosθ, u=sinθ
then u^2 s=sin^2θ(1+cosθ)
=(1-cos^2θ)(1+cosθ)
=(1+cosθ)^2(1-cosθ)
Let 1+cosθ=p, 1-cosθ=q
p^2*2q<=[(p+p+2q)/3]^3=(4/3)^3
u^2*s=p^2*q<=32/27
hence, max V=32π/81=(4π/3)(8/27)
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