Ratio of edges

The dimensions of the cuboid are in a ratio 3: 1: 2. The body diagonal has a length of 28 cm. Find the volume of a cuboid.

Result

V =  2514.394 cm³

Solution:

$u=28 \ \text{cm} \ \\ a=3x \ \\ b=1x \ \\ c=2x \ \\ u=\sqrt{ a^2+b^2+c^2 } \ \\ u^2=(9+1+4)x^2 \ \\ x=\sqrt{ \dfrac{ u^2 }{ 9+1+4 } }=\sqrt{ \dfrac{ 28^2 }{ 9+1+4 } } \doteq 2 \ \sqrt{ 14 } \ \text{m} \doteq 7.4833 \ \text{m} \ \\ a=3 \cdot \ x=3 \cdot \ 7.4833 \doteq 6 \ \sqrt{ 14 } \ \text{m} \doteq 22.4499 \ \text{m} \ \\ b=x=7.4833 \doteq 2 \ \sqrt{ 14 } \ \text{m} \doteq 7.4833 \ \text{m} \ \\ c=2 \cdot \ x=2 \cdot \ 7.4833 \doteq 4 \ \sqrt{ 14 } \ \text{m} \doteq 14.9666 \ \text{m} \ \\ \ \\ V=a \cdot \ b \cdot \ c=22.4499 \cdot \ 7.4833 \cdot \ 14.9666 \doteq 672 \ \sqrt{ 14 } \doteq 2514.3938 \doteq 2514.394 \ \text{cm}^3$

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