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The area of 3 adjacent faces of a cuboid are in the ratio of $2:3:4$ and the volume is $9000\ cm^3$, find the shortestt side.
Given:
The area of 3 adjacent faces of cuboid are in ratio of $2:3:4$ and its volume is $900\ cm^3$.
To do:
We have to find the length of the shortest side.
Solution :
Let the edges of the cuboid be $a,b,c$.
The area of the three faces of the cuboid is in the ratio $2:3:4$.
This implies,
$ab:bc:ac=2:3:4$
Let $ab=2k , bc=3k$ and $ca=4k$
Therefore,
$ab \times bc \times ca=2k \times 3k \times 4k$
$a^2b^2c^2=24k^3$
$(abc)^2=24k^3$......(i)
Volume of the cuboid $=abc$
This implies,
$(9000)^2=24k^3$
$81000000=24k^3$
$\Rightarrow k^3=\frac{81000000}{24}$
$\Rightarrow k^3=3375000$
$\Rightarrow k^3=(150)^3$
$\Rightarrow k=150$
Therefore,
$ab=2k=2(150)=300$
$bc=3k=3(150)=450$
$ca=4k=4(150)=600$
This implies,
$abc=9000$
$300c=9000$
$c=30$
Similarly,
$450a=9000$
$a=20$
$600b=9000$
$b=15$
Therefore, the length of the shortest side is $15\ cm$.