Question

Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube?

Answer 1

Hint:
 First we find the relation between the cuboids and the cube. We try to equate their area. The total area of the cuboids is equal to the area of the cube. Then we try to form the criteria for the variable that we assumed for the number of cuboids.

Complete step by step answer:
We know that the main criteria of a cube are that all the sides of a cube are equal in length.
Let us assume that Parikshit needs x number of cuboids of plasticine of sides 5 cm, 2 cm, 5 cm to make a cube.
The length and the breadth of the cuboid are equal and greater than the height of the cuboid.
Let all the sides of the cube is cm.

So, length and breadth wise we need $ \dfrac{a}{5} $ number of cuboids, and height wise we need $ \dfrac{a}{2} $ number of cuboids.
The multiplication these two numbers should be x.
We also know that volume of these cuboids in total would be equal to the volume of the cube.
So, volume of 1 cuboid is $ 5\times 2\times 5=50\text{ c}{{\text{m}}^{3}} $ . For x number of cuboids, the volume will be $ 50\times x=50x\text{ c}{{\text{m}}^{3}} $ .
The volume of the cube is $ {{a}^{3}}\text{ c}{{\text{m}}^{3}} $ which gives $ {{a}^{3}}=50x $ .
So, value of 50x has to be a cube value. We also need to find the minimum value for x.
We get that 50 has two 5s and one 2. To make 50x a cube form we need two 2s and one 5.
So, value of x becomes $ x=2\times 2\times 5=20 $ .
The number of cuboids required is 20.
Note:
 We only had one equation to solve the two-variable problem. The solution is more conceptual than being calculative. We need to form a cube where all the sides are equal and we also need to find the number x as an integer.