Question

3 cubes of metal whose sides are in ratio 3:4:5 are melted and converted into a single cube whose diagonal is 18√3 cm. Find sides of cubes.

Answer 1

Hint: Here, We are going to calculate the volume of each cube. Then use the diagonal and side relation of the cube to determine the side of the single cube. Then equate the volume of three cubes and the single cube.

Complete step-by-step answer:
Given:
The ratio of the sides of three cubes is 3: 4: 5.
The diagonal of a single cube is 18√3 cm.
Let’s assume the side of cubes be x.
Such that the volume of each cube will be as follows, ${{\left( 3x \right)}^{3}}$ ,${{\left( 4x \right)}^{3}}$, and ${{\left( 5x \right)}^{3}}$
Now let’s consider the side of a single cube be, a.
Now we have to determine the side of a single cube using the below expression.
$a=\dfrac{d}{\sqrt{3}}$
In the above expression a is the side of the cube, and d is the diagonal of the cube.
Substitute the value of variable d in the above expression.
$
   a=\dfrac{18\sqrt{3}\ \text{cm}}{\sqrt{3}} \\
  =18\ \text{cm}
$
Determine the volume of a single cube as follows.
${{\left( 18 \right)}^{3}}$ = $5832\ \text{c}{{\text{m}}^{\text{3}}}$
Now we have to equate the volume of 3 cube and the volume of a single cube as follows.
$
{{\left( 3x \right)}^{3}}+{{\left( 4x \right)}^{3}}+{{\left( 5x \right)}^{3}}=5832\ \text{c}{{\text{m}}^{\text{3}}} \\
  216{{x}^{3}}=5832\ \text{c}{{\text{m}}^{\text{3}}} \\
  {{x}^{3}}=27\ \text{c}{{\text{m}}^{\text{3}}} \\
  x=3\ \text{cm}
$
Therefore, the side of cubes will be 3 cm.
So, the correct answer is “3 cm”.

Note: We have to get the knowledge on relation of diagonal and side of the cube, because it is rarely used expression. Also we have to learn about the cube and cube roots so that we will not make mistakes. We have to be aware about the concepts on ratios so that because it is used in this question. Therefore, we have to take all the above concepts while solving the above question.