Volume ratio

Calculate the volume ratio of balls circumscribed (diameter r) and inscribed (diameter ϱ) into an equilateral rotating cone.

Correct result:

r =  8:1

Solution:

a=1  r1=3/3 a=3/3 10.5774 r2=3/6 a=3/6 10.2887   V1=43 π r13=43 3.1416 0.577430.8061 V2=43 π r23=43 3.1416 0.288730.1008  r=V1V2=0.80610.10088=8:1a=1 \ \\ \ \\ r_{1}=\sqrt{ 3 }/3 \cdot \ a=\sqrt{ 3 }/3 \cdot \ 1 \doteq 0.5774 \ \\ r_{2}=\sqrt{ 3 }/6 \cdot \ a=\sqrt{ 3 }/6 \cdot \ 1 \doteq 0.2887 \ \\ \ \\ \ \\ V_{1}=\dfrac{ 4 }{ 3 } \cdot \ \pi \cdot \ r_{1}^3=\dfrac{ 4 }{ 3 } \cdot \ 3.1416 \cdot \ 0.5774^3 \doteq 0.8061 \ \\ V_{2}=\dfrac{ 4 }{ 3 } \cdot \ \pi \cdot \ r_{2}^3=\dfrac{ 4 }{ 3 } \cdot \ 3.1416 \cdot \ 0.2887^3 \doteq 0.1008 \ \\ \ \\ r=\dfrac{ V_{1} }{ V_{2} }=\dfrac{ 0.8061 }{ 0.1008 }≈ 8=8:1



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