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.

Result

V =  1500 dm3

Solution:

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

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