The dimensions of a cuboid are in the ratio 5 : 3 : 1 and its total surface area is 414 m

It is given that the sides of the cuboid are in the ratio $5: 3: 1$.

Suppose that its sides are $x$ multiple of each other, then we have:

Length $=5 x \mathrm{~m}$

Breadth $=3 x \mathrm{~m}$

Height $=x \mathrm{~m}$

Also, total surface area of the cuboid $=414 \mathrm{~m}^{2}$

Surface area of the cuboid $=2 \times($ length $\times$ breadth $+$ breadth $\times$ height $+$ length $\times$ height $)$

$\Rightarrow 414=2 \times(5 x \times 3 x+3 x \times 1 x+5 x \times x)$

$\Rightarrow 414=2 \times\left(15 x^{2}+3 x^{2}+5 x^{2}\right)$

$\Rightarrow 414=2 \times\left(23 x^{2}\right)$

$\Rightarrow 2 \times\left(23 \times x^{2}\right)=414$

$\Rightarrow\left(23 \times x^{2}\right)=\frac{414}{2}=207$

$\Rightarrow x^{2}=\frac{207}{23}=9$

$\Rightarrow x=\sqrt{9}=3$

Therefore, we have the following:

Lenght of the cuboid $=5 \times \mathrm{x}=5 \times 3=15 \mathrm{~m}$

Breadth of the cuboid $=3 \times \mathrm{x}=3 \times 3=9 \mathrm{~m}$

Height of the cuboid $=\mathrm{x}=1 \times 3=3 \mathrm{~m}$