Question

The total surface area of a cuboid is 392 cm² and the length of the cuboid is 12 cm. If the ratio of its breadth and height is 8 : 5, then the volume of the cuboid is _________.

Answer 1

Length of the cuboid, l = 12 cm

Let the breadth and height of the cuboid be b and h, respectively.

Breadth of the cuboid, b = 8x

Height of the cuboid, h = 5x

It is given that, the total surface area of cuboid is 392 cm².

∴ 2(lb + bh + hl) = 392 cm²

$\Rightarrow 2\left(12\times 8x+8x\times 5x+5x\times 12\right)=392$

$\Rightarrow 40x^2+156x-196=0$

$\Rightarrow 10x^2+39x-49=0$

$\Rightarrow 10x^2-10x+49x-49=0$

$\Rightarrow 10x\left(x-1\right)+49\left(x-1\right)=0$

$\Rightarrow \left(x-1\right)\left(10x+49\right)=0$

$\Rightarrow x-1=0\mathrm{or}10x+49=0$

$\Rightarrow x=1\mathrm{or}x=-\frac{49}{10}$

Now, x cannot be negative. So, x = 1 cm.

So,

Breadth of the cuboid, b = 8x = 8 × 1 = 8 cm

Height of the cuboid, h = 5x = 5 ×​ 1 = 5 cm

∴ Volume of the cuboid = l × b × h = 12 × 8 × 5 = 480 cm³

Thus, the volume of the cuboid is 480 cm³.

The total surface area of a cuboid is 392 cm² and the length of the cuboid is 12 cm. If the ratio of its breadth and height is 8 : 5, then the volume of the cuboid is $\underline{480{\mathrm{cm}}^3}$.