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 answer:

k₁ =  1.5
k₂ =  1.5

Step-by-step explanation:

$$\begin{gathered} D=h \\ r=h/2 \\ {V}_1=\pi r^{2}h=2\pi r^3 \\ {V}_2=\frac{4}{3}\pi r^3 \\ {k}_1=V_1/V_2=\frac{2\pi r^3}{\frac{4}{3}\pi r^3} \\ {k}_1=\frac{2}{\frac{4}{3}}=\frac{3}{2}=1\frac{1}{2}=1.5 \end{gathered}$$

$$\begin{gathered} S_1=2\pi r^2+2\pi rh=2\pi r^2+4\pi r^2 \\ {S}_1=6\pi r^2 \\ {S}_2=4\pi r^2 \\ {k}_2=S_1/S_2=\frac{6\pi r^2}{4\pi r^2} \\ {k}_2=\frac{6}{4}=\frac{3}{2}=1\frac{1}{2}=1.5 \end{gathered}$$

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