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.

Correct result:

k₁ =  1.5
k₂ =  1.5

Solution:

$$\begin{gathered} D=h \\ r=h\mathrm{/}2 \\ {V}_1=\pi r^{2}h=2\pi r^3 \\ {V}_2={\displaystyle \frac{4}{3}}\pi r^3 \\ {k}_1=V_1\mathrm{/}{V}_2={\displaystyle \frac{2\pi r^3}{{\displaystyle \frac{4}{3}}\pi r^3}} \\ {k}_1={\displaystyle \frac{2}{{\displaystyle \frac{4}{3}}}}={\displaystyle \frac{3}{2}}=1.5 \end{gathered}$$

$$\begin{gathered} S_1=2\pi r^2+2\pi rh=2\pi r^2+4\pi r^2 \\ {S}_1=6\pi r^2 \\ {S}_2=4\pi r^2 \\ {k}_2=S_1\mathrm{/}{S}_2={\displaystyle \frac{6\pi r^2}{4\pi r^2}} \\ {k}_2={\displaystyle \frac{6}{4}}={\displaystyle \frac{3}{2}}=1.5 \end{gathered}$$

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