Ratio-cuboid

The lengths of the edges of the cuboid are in the ratio 2: 3: 6. Its body diagonal is 14 cm long. Calculate the volume and surface area of the cuboid.

Correct answer:

V =  288 cm³
S =  288 cm²

Step-by-step explanation:

$$\begin{gathered} a:b:c=2:3:6 \\ a=2x \\ b=3x \\ c=6x \\ u=14\text{ cm } \\ u=\sqrt{a^2+b^2+c^2} \\ u=\sqrt{(2x{)}^2+(3x{)}^2+(6x{)}^2} \\ u=\sqrt{4x^2+9x^2+36x^2} \\ {u}^2=4x^2+9x^2+36x^2 \\ x=u\mathrm{/}\sqrt{4+9+36}=14\mathrm{/}\sqrt{4+9+36}=2\text{ cm } \\ a=2\cdot x=2\cdot 2=4\text{ cm } \\ b=3\cdot x=3\cdot 2=6\text{ cm } \\ c=6\cdot x=6\cdot 2=12\text{ cm } \\ {u}_2=\sqrt{a^2+b^2+c^2}=\sqrt{4^2+6^2+12^2}=14 \\ V=a\cdot b\cdot c=4\cdot 6\cdot 12=288{\text{cm}}^3 \end{gathered}$$

$S=2\cdot (a\cdot b+b\cdot c+a\cdot c)=2\cdot (4\cdot 6+6\cdot 12+4\cdot 12)=288{\text{cm}}^2$

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