Question

Three cubes of metal whose edges are in the ratio of 3:4:5 are melted to form a cube whose diagonal is $6\sqrt{3}$ 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

Hint: We should know the concept and formulae of surface area and volume. Volume is the space occupied by an object. In this question, the volume of the cube is used.
As we know,
Volume of cube = $edge\times edge\times edge$ = ${{a}^{3}}$

Complete step-by-step answer:
According to question,
We have been given the edge of three cube 3x, 4x & 5x respectively.

Volume of 3x cube = ${{\left( 3x \right)}^{3}}=27{{x}^{3}}$
Volume of 4x cube = ${{\left( 4x \right)}^{3}}=64{{x}^{3}}$
Volume of 5x cube = ${{\left( 5x \right)}^{3}}=125{{x}^{3}}$
We get the total volume of three cubes = $27{{x}^{3}}+64{{x}^{3}}+125{{x}^{3}}=216{{x}^{3}}$
Now, let us take ‘a’ as the edge of the new cube so formed after melting.
Volume of new cube = volume of 3 cubes
Then we have,
$\Rightarrow {{a}^{3}}=216{{x}^{3}}$
$\Rightarrow a=\sqrt{216{{x}^{3}}}$
$\therefore $ a = 6x……..(i)
We have been given in question that the diagonal of new cube = $12\sqrt{3}$
​$\Rightarrow \sqrt{{{a}^{2}}+{{a}^{2}}+{{a}^{2}}}=6\sqrt{3}$
$\Rightarrow \sqrt{3{{a}^{2}}}=6\sqrt{3}$
  $\Rightarrow a=\dfrac{6\sqrt{3}}{\sqrt{3}}$
$\therefore a = 6$
From equation(i)
$\Rightarrow $ 6 = 6x
$\Rightarrow $ $\dfrac{6}{6}=x$
$\therefore $ x=1
The edge of three cubes are given by $3\times 1:4\times 1:5\times 1$
3 : 4 : 5
$\Rightarrow $3x = 3×1 = 3 cm
$\Rightarrow $4x = 4×1 = 4 cm
$\Rightarrow $5x = 5×1 = 5 cm
Thus, the edges of three cubes are 3cm, 4cm and 5cm respectively.

So, the correct answer is “Option D”.

Note: You should be familiar with the formulae of surface area and volume of the cube. Units are mandatory. Assign the numbers to the equations used in the whole answer. If the numbers in the under root are cancelling, cancel them otherwise put their value and solve till the end.