2x cone

Circular cone height 84 cm was cut plane parallel with base. Volume of these two small cones is the same. Calculate the height of the smaller cone.

Result

v2 =  66.671 cm

Solution:

v1=84 cm r1=1 cm V1=13 π r12 v1=13 3.1416 12 8487.9646 cm3  V2=V1/2=87.9646/243.9823 cm3 r1:v1=r2:v2  V2=13 π r22 v2 V2=13 π (r1/v1 v2)2 v2 V2=13 π r12/v12 v23   v2=3 V2/π/r12 v123=3 43.9823/3.1416/12 842366.670866.671 cm  r2=r1/v1 v2=1/84 66.67080.7937 cm  V3=13 π r22 v2=13 3.1416 0.79372 66.670843.9823 cm3 V3=V2=V1/2v_{1}=84 \ \text{cm} \ \\ r_{1}=1 \ \text{cm} \ \\ V_{1}=\dfrac{ 1 }{ 3 } \cdot \ \pi \cdot \ r_{1}^2 \cdot \ v_{1}=\dfrac{ 1 }{ 3 } \cdot \ 3.1416 \cdot \ 1^2 \cdot \ 84 \doteq 87.9646 \ \text{cm}^3 \ \\ \ \\ V_{2}=V_{1}/2=87.9646/2 \doteq 43.9823 \ \text{cm}^3 \ \\ r_{1}:v_{1}=r_{2}:v_{2} \ \\ \ \\ V_{2}=\dfrac{ 1 }{ 3 } \cdot \ \pi \cdot \ r_{2}^2 \cdot \ v_{2} \ \\ V_{2}=\dfrac{ 1 }{ 3 } \cdot \ \pi \cdot \ (r_{1}/v_{1} \cdot \ v_{2})^2 \cdot \ v_{2} \ \\ V_{2}=\dfrac{ 1 }{ 3 } \cdot \ \pi \cdot \ r_{1}^2/v_{1}^2 \cdot \ v_{2}^3 \ \\ \ \\ \ \\ v_{2}=\sqrt[3]{ 3 \cdot \ V_{2} / \pi /r_{1}^2 \cdot \ v_{1}^2}=\sqrt[3]{ 3 \cdot \ 43.9823 / 3.1416 /1^2 \cdot \ 84^2} \doteq 66.6708 \doteq 66.671 \ \text{cm} \ \\ \ \\ r_{2}=r_{1}/v_{1} \cdot \ v_{2}=1/84 \cdot \ 66.6708 \doteq 0.7937 \ \text{cm} \ \\ \ \\ V_{3}=\dfrac{ 1 }{ 3 } \cdot \ \pi \cdot \ r_{2}^2 \cdot \ v_{2}=\dfrac{ 1 }{ 3 } \cdot \ 3.1416 \cdot \ 0.7937^2 \cdot \ 66.6708 \doteq 43.9823 \ \text{cm}^3 \ \\ V_{3}=V_{2}=V_{1}/2



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