# 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:

$a=1 \ \\ \ \\ r_{1}=\sqrt{ 3 }/3 \cdot \ a=\sqrt{ 3 }/3 \cdot \ 1 \doteq 0.5774 \ \\ r_{2}=\sqrt{ 3 }/6 \cdot \ a=\sqrt{ 3 }/6 \cdot \ 1 \doteq 0.2887 \ \\ \ \\ \ \\ V_{1}=\dfrac{ 4 }{ 3 } \cdot \ \pi \cdot \ r_{1}^3=\dfrac{ 4 }{ 3 } \cdot \ 3.1416 \cdot \ 0.5774^3 \doteq 0.8061 \ \\ V_{2}=\dfrac{ 4 }{ 3 } \cdot \ \pi \cdot \ r_{2}^3=\dfrac{ 4 }{ 3 } \cdot \ 3.1416 \cdot \ 0.2887^3 \doteq 0.1008 \ \\ \ \\ r=\dfrac{ V_{1} }{ V_{2} }=\dfrac{ 0.8061 }{ 0.1008 }≈ 8=8:1$

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