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 $15\ cm$.

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Updated on: 10-Oct-2022

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