Question

Harshit makes a cuboid with plasticine of sides 3cm , 2cm and 3cm. How many such cuboids are needed to form a cube
with explanation​

Answer 1

Answer:

We need 12 cuboids to make the smallest cube with side 6 cm.

Step-by-step explanation:

We have,

A cuboid of

L = 3 cm

B = 2 cm

H = 3 cm

Let x cuboids be needed to make a cube

Now,

Since we are taking 'x' number of cuboids and then making them into a cube, we need some value that is constant for both cuboid and cube, only then we can equate them.

Here, when we check,

Their sides will be different.

Surface Area could be same or different (we are not sure so we can't take it either).

But if we observe carefully their Volume will be equal.

No matter how many cuboids we take their total volume must be equal to the volume of a cube if they are turned into a cube.

Then,

Volume of cuboid = L × B × H

V = 3 × 2 × 3

V = 3² × 2

We don't need to multiply and find the answer you will find out why, in the later solution.

Now, we are using x cuboids

So,

Total Volume = x × Volume of 1 cuboid

Total Volume = x × 3² × 2

Also,

Let the side of the cube be a

So,

Volume of a cube with side a will be,

V = a³

Then,

Volume of cube = Volume of x cuboids

So,

a³ = 3² × 2 × x

a = ³√(3² × 2 × x)

Now,

'a' must be a whole number because it is the side of a cube formed from a cuboid which has a whole number as it's side.

So,

√(3² × 2 × x) must be a whole number

which means

(3² × 2 × x) must be a perfect cube (number).

Now, we must know that,

A perfect cube is a number with prime factorization of product of primes with powers of 3 or multiples of 3.

Here,

3² has power 2, we need it to be 3

So,

we should just multiply 3

So,

3² × 3 = 3³

Similarly,

2 has a power 1, we need it to be 3

So,

we should just multiply by 2²

So,

2 × 2² = 2³

Which means value of x must be (3 × 2²)

Then only it will make it a perfect cube right?!. Think about it....!!!

See,

3² × 2 × x

= 3² × 2 × 3 × 2²

= 3³ × 2³

and √(3³ × 2³) = 3 × 2 = 6

So, we get that,

x = 3 × 2²

x = 3 × 4

x = 12

Which means,

We need 12 cuboids to make the smallest cube with side 6 cm.

So, see if we had multiplied the volume directly, we wouldn't have been able to factorise with primes. At that stage we should have found out the primes and then write it which will take more time and make the Question tiring.

We could make bigger cube like we want with more number of cuboids but we only need the smallest cube that can be formed, so only 12 cuboid is needed.

Hope it helped you and believing you understood it...All the best