How many balls of radii 1 cm can be made by melting a cube of side 22 cm?

Options

A. 5324
B. 2662
C. 2541
D. 1347

Answer

To find out how many balls of radius 1 cm can be made by melting a cube of side 22 cm, we will follow these steps:

### Step 1: Calculate the Volume of the Cube
The formula for the volume \( V \) of a cube with side length \( a \) is given by:
\[ V = a^3 \]
For a cube with side length 22 cm:
\[ V = 22^3 = 22 \times 22 \times 22 = 10648 \text{ cm}^3 \]

### Step 2: Calculate the Volume of One Ball
The formula for the volume \( V \) of a sphere with radius \( r \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For a ball with radius 1 cm:
\[ V = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \text{ cm}^3 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{4}{3} \times \frac{22}{7} \times 1 = \frac{88}{21} \text{ cm}^3 \]

### Step 3: Calculate the Number of Balls
To find the number of balls that can be made, we divide the volume of the cube by the volume of one ball:
\[ \text{Number of balls} = \frac{\text{Volume of cube}}{\text{Volume of one ball}} = \frac{10648}{\frac{88}{21}} \]
This simplifies to:
\[ \text{Number of balls} = 10648 \times \frac{21}{88} \]
Calculating this gives:
\[ \text{Number of balls} = \frac{10648 \times 21}{88} \]
Calculating \( 10648 \times 21 = 223608 \) and then dividing by 88:
\[ \text{Number of balls} = \frac{223608}{88} = 2536 \]

### Final Answer
Thus, the number of balls of radius 1 cm that can be made by melting a cube of side 22 cm is **2536**.

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