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

Answer: 540

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 $=Ibh=12\times 9\times 5=540{\mathrm{cm}}^3$