# Right circular cone

The volume of a right circular cone is 5 liters. Calculate the volume of the two parts into which the cone is divided by a plane parallel to the base, one-third of the way down from the vertex to the base.

Result

V1 =  0.185 l
V2 =  4.815 l

#### Solution:

$V = 5 \ l = 5 \cdot \ 1 \ dm^3 = 5 \ dm^3 \ \\ \ \\ V = \dfrac{ 1 }{ 3 } \cdot \ S \cdot \ h \ \\ \ \\ h_{ 1 } = \dfrac{ 1 }{ 3 } h \ \\ S_{ 1 } = \dfrac{ 1 }{ 3^2 } S = \dfrac{ 1 }{ 9 } S \ \\ \ \\ V_{ 1 } = \dfrac{ 1 }{ 3 } \cdot \ S_{ 1 } \cdot \ h_{ 1 } = \dfrac{ 1 }{ 3 } \cdot \ \dfrac{ 1 }{ 9 } S \cdot \ \dfrac{ 1 }{ 3 } h \ \\ \ \\ V_{ 1 } = V \cdot \ \dfrac{ 1 }{ 3^3 } = 5 \cdot \ \dfrac{ 1 }{ 3^3 } = \dfrac{ 5 }{ 27 } \doteq 0.1852 = 0.185 \ \text{ l }$
$V_{ 2 } = V-V_{ 1 } = 5-0.1852 = \dfrac{ 963 }{ 200 } = 4.815 = 4.815 \ \text{ l }$

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Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 1 comment:
Dr Math
1:3 is the ratio of heights
1:32 = 1:9 is the ratio of the area of base circles... due to two dimensional nature of the area.
1:33 = 1:27 is the ratio of volumes...  .. volume has three-dimensional nature

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