Question

The length of the longest diagonal of a cuboid is $5 \sqrt{10} \mathrm{~cm}$, and the length of one of the longest face diagonals is $15 \mathrm{~cm}$. If the lengths of the edges are integers in $\mathrm{cm}$, then the volume of the cuboid (in $\mathrm{cm}^3$ ) is___

Answer 1

Let $l, b$, and $h$ be the length, breadth, and height of the cuboid, respectively.
The longest diagonal of the cuboid
$ =\sqrt{l^2+b^2+h^2}=5 \sqrt{10} \mathrm{~cm}$
$ \Rightarrow l^2+b^2+h^2=250$
$ \text { and } \sqrt{l^2+b^2}=15$
( $\because$ One of the longest face diagonals)
$\Rightarrow l^2+b^2=225 \Rightarrow l=12 \text { and } b=9$
( $\because$ The lengths of the edges are integers.)
$\therefore l^2+b^2+h^2 =250 $
$225+h^2 =250$
$h^2 =25$
$h =5 \mathrm{~cm}$
$\therefore$ Volume $=I b h=12 \times 9 \times 5=540 \mathrm{~cm}^3$