The dimensions of a cuboid are in the ratio $1 : 2 : 3$ and its volume is $384 c u . m$ . Find the dimensions and surface area.

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Solution

Dimensions of the cuboid $= l : b : h = x : 2 x : 3 x$

Volume of the cuboid $= l b h = 384 c u . m$

$\Rightarrow x \times 2 x \times 3 x = 384$

$\Rightarrow 6 x^{3} = 384$

$\Rightarrow x^{3} = \frac{384}{6} = 64$

$\Rightarrow x = 4 m$

$\Rightarrow$ length $= l = x = 4 m$

Breadth $= b = 2 x = 2 \times 4 = 8 m$ and height $= h = 3 x = 3 \times 4 = 12 m$

Thus, dimensions of the cuboid $= 4 : 8 : 12$

Total surface area $= 2 (l b + b h + l h)$

$= 2 (4 \times 8 + 8 \times 12 + 4 \times 12)$

$= 2 (32 + 96 + 48)$

$= 2 \times 176 = 352 s q . m$

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