Question

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

Answer 1

Hint: First, express the edges of ratio in quantities like 3x, 4x, 5x then find the volume of each and add it using the formula, ${{a}^{3}}$, where a is the edge length of the cube. The sum will be the volume of the bigger cube. As the diagonal of the formed cube is given, an edge can be found using the formula, $a\sqrt{3}$, where it represents it’s edge length. Then, find the volume and compare it with the sum of volumes to get the value of x and edges.

Complete step-by-step answer:
In the question, we are given the ratio of edges of three cubes which are in the ratio 3:4:5. Then, it is said that they are melted and formed into a single cube. The diagonal of the formed single cube is $12\sqrt{3}cm$. Now let the edge of cubes be represented as 3x, 4x and 5x, where x is a constant which when multiplied gives the length of the edges of the respective cubes.
The diagonal of the new cube is $12\sqrt{3}cm$. As we know that the formula of the diagonal of the cube is $a\sqrt{3}$, where it represents it’s edge length. So, we get, $a\sqrt{3}=12\sqrt{3}$. By cancelling the like terms, we get, a = 12 cm. So, 12 cm is the edge length of the new cube.
Now as we know that all the three cubes were melted to form a new cube, we get the sum of volumes of the three cubes equal to the volume of the larger cube. So, to find the volume of the cube, we can use the formula, ${{a}^{3}}$, where a is the edge length of the cube.
Hence, the volume of the cube with the edge 3x is ${{(3x)}^{3}}=27{{x}^{3}}$.
The volume of the cube with the edge 4x is ${{(4x)}^{3}}=64{{x}^{3}}$.
The volume of the cube with the edge 5x is ${{(5x)}^{3}}=125{{x}^{3}}$.
So, the sum of the volumes of three cubes is, $27{{x}^{3}}+64{{x}^{3}}+125{{x}^{3}}=216{{x}^{3}}$.
The volume of the formed cube of edge length 12 cm is ${{(12cm)}^{3}}=1728c{{m}^{3}}$.
So, we can say that $216{{x}^{3}}=1728$. So, on simplifying we get, ${{x}^{3}}=8$, hence we get $x=2$.
So the edges of the three cubes are $\left( 3\times 2 \right)cm,\left( 4\times 2 \right)cm,\left( 5\times 2 \right)cm$, which is 6 cm, 8 cm and 10 cm respectively.
Hence the edges of the three cubes are 6 cm, 8 cm and 10 cm.

Note: Students should know how to use the concept of volume and surface area in these types of questions which can only be achieved through practice. The key concept in this question is the sum of the volume of three metal cubes will be equal to a bigger cube.