Cuboid and ratio

Cuboid has dimensions in ratio 1:2:6 and the surface area of the cuboid is 1000 dm2. Calculate the volume of the cuboid.

Correct result:

V =  1500 dm3

Solution:

V=abc a:b:c=1:2:6 a=1k b=2k c=6k S=1000 S=2(ab+bc+ac) S=2(2k2+12k2+6k2)=40 k2 k=S/40=1000/40=5 a=k=5 b=2⋅ k=2⋅ 5=10 c=6⋅ k=6⋅ 5=30 V=a⋅ b⋅ c=5⋅ 10⋅ 30=1500 dm3V=abc \ \\ a:b:c=1:2:6 \ \\ a=1k \ \\ b=2k \ \\ c=6k \ \\ S=1000 \ \\ S=2(ab+bc+ac) \ \\ S=2(2k^2+12k^2+6k^2)=40 \ k^2 \ \\ k=\sqrt{ S/ 40 }=\sqrt{ 1000/ 40 }=5 \ \\ a=k=5 \ \\ b=2 \cdot \ k=2 \cdot \ 5=10 \ \\ c=6 \cdot \ k=6 \cdot \ 5=30 \ \\ V=a \cdot \ b \cdot \ c=5 \cdot \ 10 \cdot \ 30=1500 \ \text{dm}^3

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