Question

The length, breadth and height of a cuboid are in the ratio $2:3:2\sqrt{3}$. Then, the ratio of the diagonal to the height of the cuboid is

• A. $5:\sqrt{3}$
• B. $5:2\sqrt{3}$
• C. $10:3\sqrt{3}$
• D. $4:\sqrt{3}$

Answer 1

The correct option is B $5:2\sqrt{3}$

Let the length, breadth and height of the cuboid be $l$, $b$ and $h$, respectively.
$\therefore l:b:h=2:3:2\sqrt{3}$ (Given)
$\Rightarrow l=2k,b=3k,h=2\sqrt{3}k,$ where $k$ is any natural number
Diagonal of cuboid(d) is given by
$d=\sqrt{1^2+b^2+h^2}$
$=\sqrt{\left(2k^2\right)+(3k{)}^2+(2\sqrt{3}k{)}^2}$
$=\sqrt{4k^2+9k^2+12k^2}$
$=\sqrt{25k^2}$
= 5$k$
∴ Ratio of the diagonal to the height of the cuboid = $d:h$
$=5k:2\sqrt{3}k$
$=5:2\sqrt{3}$
Hence, the correct answer is option (b).