Question

Three cubes with sides in the ratio $3:4:5$ are melted to form a single cube whose diagonal is $12\sqrt{3}cm$. The sides of the cubes are

• A. $3cm,4cm,5cm$
• B. $6cm,8cm,10cm$
• C. $9cm,12cm,15cm$
• D. None of these

Answer 1

Answer: (B)

$6cm,8cm,10cm$

Let the sides of the cube be $3a,4a,5a$ respectively
Length
of the diagonal of a cube $=\sqrt{3}\times side=12\sqrt{3}$
=> Side of the larger cube $=12cm$
Volume
of resulting cube $=$ Volume of cube $1+$ Volume of
cube $2+$ Volume of cube $3$
Volume of a cube of edge a $=a^3$
So,

${12}^3={\left(3a\right)}^3+{\left(4a\right)}^3+{\left(5a\right)}^3$
$ => 1728 = 27{a}^{3} + 64{a}^{3} + 125{a}^{3} => 216{ a }^{ 3 } = 1728 $

$ { a }^{ 3 } = 8 $

$=>a=2$
Hence, length of sides of the cubes are $6cm,8cm,10cm$