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?

Options

A. `22`
B. `21`
C. `23`
D. `20`

Explanation

To solve the problem of how many cuboids Parikshit needs to form a cube, we will follow these steps:

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is calculated using the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
For the given cuboid with dimensions 5 cm, 2 cm, and 5 cm:
\[ V = 5 \, \text{cm} \times 2 \, \text{cm} \times 5 \, \text{cm} \]
Calculating this gives:
\[ V = 5 \times 2 \times 5 = 50 \, \text{cm}^3 \]

### Step 2: Determine the Volume of the Cube
To form a cube, we need to find a cube whose volume is a multiple of the volume of the cuboid. The volume \( V_c \) of a cube is given by:
\[ V_c = a^3 \]
where \( a \) is the length of a side of the cube.

### Step 3: Find the Smallest Cube Volume Greater than the Cuboid Volume
The smallest cube that can be formed must have a volume that is a multiple of 50 cm³. We can start with the smallest integer \( n \) such that \( n^3 \) is greater than or equal to 50 cm³.

Calculating the cubes of integers:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)

The smallest cube that is greater than 50 cm³ is \( 4^3 = 64 \, \text{cm}^3 \).

### Step 4: Calculate the Number of Cuboids Needed
Now, to find the number of cuboids needed to form this cube, we divide the volume of the cube by the volume of the cuboid:
\[ \text{Number of cuboids} = \frac{V_c}{V} = \frac{64 \, \text{cm}^3}{50 \, \text{cm}^3} \]
Calculating this gives:
\[ \text{Number of cuboids} = \frac{64}{50} = 1.28 \]

Since we cannot have a fraction of a cuboid, we round up to the nearest whole number. Thus, we need 2 cuboids to form a cube of volume 64 cm³.

### Final Answer
Parikshit will need **2 cuboids** to form a cube.

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