Volume of three cuboids

Calculate the total volume of all cuboids for which the the size of the edges are in a ratio of 1:2:3, and one of the edges has a size 6 cm.

Result

V =  1506 cm3

Solution:

x=6 a1=x=6 b1=2x=2⋅ 6=12 c1=3x=3⋅ 6=18 V1=a1⋅ b1⋅ c1=6⋅ 12⋅ 18=1296 a2=x/2=6/2=3 b2=x=6 c2=3⋅ x/2=3⋅ 6/2=9 V2=a2⋅ b2⋅ c2=3⋅ 6⋅ 9=162 a3=x/3=6/3=2 b3=2⋅ x/3=2⋅ 6/3=4 c3=x=6 V3=a3⋅ b3⋅ c3=2⋅ 4⋅ 6=48 V=V1+V2+V3=1296+162+48=1506=1506 cm3x = 6 \ \\ a_{ 1 } = x = 6 \ \\ b_{ 1 } = 2x = 2 \cdot \ 6 = 12 \ \\ c_{ 1 } = 3x = 3 \cdot \ 6 = 18 \ \\ V_{ 1 } = a_{ 1 } \cdot \ b_{ 1 } \cdot \ c_{ 1 } = 6 \cdot \ 12 \cdot \ 18 = 1296 \ \\ a_{ 2 } = x/2 = 6/2 = 3 \ \\ b_{ 2 } = x = 6 \ \\ c_{ 2 } = 3 \cdot \ x/2 = 3 \cdot \ 6/2 = 9 \ \\ V_{ 2 } = a_{ 2 } \cdot \ b_{ 2 } \cdot \ c_{ 2 } = 3 \cdot \ 6 \cdot \ 9 = 162 \ \\ a_{ 3 } = x/3 = 6/3 = 2 \ \\ b_{ 3 } = 2 \cdot \ x/3 = 2 \cdot \ 6/3 = 4 \ \\ c_{ 3 } = x = 6 \ \\ V_{ 3 } = a_{ 3 } \cdot \ b_{ 3 } \cdot \ c_{ 3 } = 2 \cdot \ 4 \cdot \ 6 = 48 \ \\ V = V_{ 1 }+V_{ 2 }+V_{ 3 } = 1296+162+48 = 1506 = 1506 \ cm^3

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