Question

Three cubes of metal whose edges are in the ratio $3:4:5$ are melted to form a cube whose diagonal is $12\sqrt{3}$ cm. Find the edges (in cm) of the three cubes.

• A. $6,8,10$
• B. $9,12,15$
• C. $8,9,11$
• D. $3,4,5$

Answer 1

Answer: (A)

Given:

The edges of cubes are in ratio of $=3:4:5$

Let the edges be $3x,4x$ & $5x$ respectively .

Since, the volume of cube $=a^3$ cubic units, [$a$ is edge of the cube.]

$\Rightarrow$ Volume of the cube with side $\left(3x\right)=(3x{)}^3=27x^3$

Volume of the cube with side $\left(4x\right)=(4x{)}^3=64x^3$

Volume of the cube with side $\left(5x\right)=(5x{)}^3=125x^3$

Volume of the cube formed after melting the $3$ cubes $=27x^3+64x^3+125x^3=216x^3$

Let the side of the new cube be $l$.

$\Rightarrow l^3=216x^3$

$\Rightarrow l^3=(6x{)}^3$

$\Rightarrow l=6x$

i.e., side of the new cube $\left(l\right)=6x$.

Since, if side of the cube is $l$, the its length of the diagonal is

$\left(d\right)=\sqrt{3}l$ units [Observe the derivation]

It is given that, diagonal of new cube $=12\sqrt{3}$

$\Rightarrow \sqrt{3}l=12\sqrt{3}$

$\Rightarrow l=12$

$\Rightarrow 6x=12$

$\Rightarrow x=\frac{12}{6}$

$\Rightarrow x=2$

Hence, the edges of the cubes are,

$3x=3\times 2=6$

$4x=4\times 2=8$

$5x=5\times 2=10$

Hence, Option A is correct.

Derivation of length of a diagonal of a cube:

Let the edge of the cube be $l$ units.

Since, the cube is bounded by $6$ congruent squares.

In figure, length of the diagonal is $d$ units.

In $△DHG,\angle DHG={90}^{\circ }$

By Pythagoras theorem,

$D{G}^2=D{H}^2+H{G}^2$

$\Rightarrow D{G}^2=l^2+l^2$

$\Rightarrow D{G}^2=2l^2$ ---(1)

Now, in $△ADG,\angle ADG={90}^{\circ }$

$\Rightarrow A{G}^2=A{D}^2+D{G}^2$

$\Rightarrow d^2=l^2+2l^2$

$\Rightarrow d^2=3l^2$

$\Rightarrow d=\sqrt{3}l$.

Hence, the diagonal of the cube is $\left(d\right)=\sqrt{3}l$ units