Question

The ratio of the length, breadth, and height of a cuboid is 5: 3: 2. If the total surface area of the cuboid in 558 cm${}^2$. Find the length of its sides.

Answer 1

Given that,

The ratio of the length, breadth, and height of a cuboid is $5:3:2$.

The total surface area of the cuboid is $558c{m}^2$.

To find out,

The length of the sides of the cuboid.

Let the length of the cuboid be $5xcm$.

Hence, its breadth and height will be $3xcm$ and $2xcm$ respectively.

We know that, the surface area of a cuboid $=2(lb+bh+hl)$

Hence, $2(lb+bh+hl)=558$

$\Rightarrow 2(5x\times 3x+3x\times 2x+2x\times 5x)=558$

$\Rightarrow 2(15x^2+6x^2+10x^2)=558$

$\Rightarrow 2\left(31x^2\right)=558$

$\Rightarrow 62x^2=558$

$\Rightarrow x^2=\frac{558}{62}$

$\Rightarrow x^2=9$

$\Rightarrow x=\pm 3$

As $5x,3x,2x$ are length, breadth and height of a cuboid, $x$ cannot be negative.

$\therefore x=3$

Hence, length $=5x=5\times 3$

$=15cm$

breadth $=3x=3\times 3$

$=9cm$

And height $=2x=2\times 3$

$=6cm$

Hence, the length, breadth and height of the cuboid are $15cm,9cm$ and $6cm$ respectively.