Question

The total surface area of a cuboid is $392{cm}^2$ and the length of the cuboid is $12cm$. If the ratio of its breadth and height is $8:5$ then the volume of the cuboid is .

Answer 1

Given: Total surface area of a cuboid $=392{cm}^2$
Length of the cuboid $=12cm$
Let the length, breadth and height of the cuboid be $l,b$ and $h$ respectively such that $b=8x$ and $h=5x$

Since, total surface area of cuboid $=392{cm}^2$
$2(lb+bh+hl)=392$
$2(12\times 8x+8x\times 5x+5x\times 12)=392$
$96x+40x^2+60x=196$
$24x+10x^2+15x=49$
$10x^2+39x-49=0$
$10x^2-10x+49x-49=0$
$10x(x-1)+49(x-1)=0$
$(10x+49)(x-1)=0$
$x=1;x=-\frac{49}{10}$
Since, $x$ cannot be negative
Thus, $x=1cm$

$\therefore$ Breadth of the cuboid, $b=8x=8\times 1=8cm$
Height of the cuboid, $h=5x=5\times 1=5cm$
$\therefore$ Volume of the cuboid $=l\times b\times h$
$=12\times 8\times 5=480$
$=480{cm}^3$
Thus, the volume of the cuboid is $480{cm}^3$