Three cubes of a metal whose edges are in the ratio $ 3: 4: 5 $ are melted and converted into a single cube whose diagonal is $ 12 \sqrt{3} \mathrm{~cm} $. Find the edges of the three cubes.

Given:

Three cubes of a metal whose edges are in the ratio \( 3: 4: 5 \) are melted and converted into a single cube whose diagonal is \( 12 \sqrt{3} \mathrm{~cm} \). 

To do:

We have to find the edges of the three cubes.

Solution:

Let the edges of the three cubes be $3x, 4x$ and $5x$ respectively and $a$ be the length of the side of the cube formed after melting.

Volume of the cubes after melting is $= (3x)^3+ (4x)^3+ (5x)^3$

$= 27x^3+64x^3+125x^3$

$=216x^3$

Therefore,

$a^3= 216x^3$

$a^3 = (6x)^3$

$\Rightarrow a=6x$

Diagonal of the cube $=\sqrt{3} a$

This implies,

$\sqrt{3} a=12 \sqrt{3}$

$a=12$

$a=6x=12$

$x=2$

$\Rightarrow 3x=3(2)=6\ cm$

$\Rightarrow 4x=4(2)=8\ cm$

$\Rightarrow 5x=5(2)=10\ cm$

Therefore, the edges of the three cubes are $6 \mathrm{cm}, 8 \mathrm{~cm}$ and $10 \mathrm{~cm}$ respectively.

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Updated on: 10-Oct-2022

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