Cuboid - ratios

The sizes of the edges of the cuboid are in the ratio 2: 3: 5. The smallest wall have area 54 cm2. Calculate the surface area and volume of this cuboid.

Result

S =  558 cm2
V =  810 cm3

Solution:

$a=2x \ \\ b=3x \ \\ c=5x \ \\ ab=54 \ \\ 6x^2=54 \ \\ x=\sqrt{ 54/6 }=3 \ \\ a=2 \cdot \ x=2 \cdot \ 3=6 \ \\ b=3 \cdot \ x=3 \cdot \ 3=9 \ \\ c=5 \cdot \ x=5 \cdot \ 3=15 \ \\ S_{1}=a \cdot \ b=6 \cdot \ 9=54 \ \\ S_{2}=b \cdot \ c=9 \cdot \ 15=135 \ \\ S_{3}=a \cdot \ c=6 \cdot \ 15=90 \ \\ S=2 \cdot \ (S_{1}+S_{2}+S_{3})=2 \cdot \ (54+135+90)=558 \ \text{cm}^2$
$V=a \cdot \ b \cdot \ c=6 \cdot \ 9 \cdot \ 15=810 \ \text{cm}^3$

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