Equilateral cylinder

A sphere is inserted into the rotating equilateral cylinder (touching the bases and the shell). Prove that the cylinder has both a volume and a surface half larger than an inscribed sphere.

Result

k1 =  1.5
k2 =  1.5

Solution:

D=h r=h/2 V1=πr2 h=2πr3  V2=43πr3  k1=V1/V2=2 πr343πr3  k1=243=32=1.5D=h \ \\ r=h/2 \ \\ V_{1}=\pi r^2 \ h=2\pi r^3 \ \\ \ \\ V_{2}=\dfrac{ 4 }{ 3 } \pi r^3 \ \\ \ \\ k_{1}=V_{1}/V_{2}=\dfrac{ 2 \ \pi r^3 }{ \dfrac{ 4 }{ 3 } \pi r^3 } \ \\ \ \\ k_{1}=\dfrac{ 2 }{ \dfrac{ 4 }{ 3 } }=\dfrac{ 3 }{ 2 }=1.5
S1=2 πr2+2 πrh=2 πr2+4 πr2 S1=6 πr2  S2=4 πr2  k2=S1/S2=6 πr24 πr2  k2=64=32=1.5S_{1}=2 \ \pi r^2 + 2 \ \pi r h=2 \ \pi r^2 + 4 \ \pi r^2 \ \\ S_{1}=6 \ \pi r^2 \ \\ \ \\ S_{2}=4 \ \pi r^2 \ \\ \ \\ k_{2}=S_{1}/S_{2}=\dfrac{ 6 \ \pi r^2 }{ 4 \ \pi r^2 } \ \\ \ \\ k_{2}=\dfrac{ 6 }{ 4 }=\dfrac{ 3 }{ 2 }=1.5



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