The dimensions of a cuboid are in the ratio $5 : 3 : 1$ and its total surface area is $414 m^{2}$ . Find the dimensions of the cuboid.

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Solution

Dimensions of a cuboid are in the ratio $= 5 : 3 : 1$
$∴$ length $= l = 5 x$
Breadth $= b = 3 x$ and height $= h = x$
Total surface area of a cuboid $= 2 (l b + b h + l h)$
$\Rightarrow 414 = 2 (5 x \times 3 x + 3 x \times x + 5 x \times x)$
$\Rightarrow 414 = 2 (15 x^{2} + 3 x^{2} + 5 x^{2})$
$\Rightarrow 414 = 2 \times 23 x^{2}$
$\Rightarrow 207 = 23 x^{2}$
$\Rightarrow x^{2} = \frac{207}{23} = 9$
$∴ x = \sqrt{9} = 3$
$∴$ Length $= 5 x = 5 \times 3 = 15$ m
Breadth $= 3 x = 3 \times 3 = 9$ m
and height $= h = x = 3$ m

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