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 solve the problem of 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 to calculate the volume \( V \) of a cube with side length \( a \) is given by:
\[ V = a^3 \]
For our cube with side length 22 cm:
\[ V = 22^3 = 22 \times 22 \times 22 \]
Calculating this:
\[ 22 \times 22 = 484 \]
Then,
\[ 484 \times 22 = 10648 \text{ cm}^3 \]
So, the volume of the cube is \( 10648 \text{ cm}^3 \).

### Step 2: Calculate the volume of one ball.
The formula to calculate the volume \( V \) of a sphere with radius \( r \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For our ball with radius 1 cm:
\[ V = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \text{ cm}^3 \]
Using \( \pi \approx 3.14 \):
\[ V \approx \frac{4}{3} \times 3.14 \approx \frac{12.56}{3} \approx 4.19 \text{ cm}^3 \]

### Step 3: Calculate the number of balls that can be made.
To find the number of balls that can be made, we divide the volume of the cube by the volume of one ball:
\[ \text{Number of balls} = \frac{\text{Volume of cube}}{\text{Volume of one ball}} = \frac{10648}{\frac{4}{3} \pi} \]
Substituting the approximate value of \( \pi \):
\[ \text{Number of balls} \approx \frac{10648}{4.19} \approx 2546.61 \]
Since we can only have whole balls, we take the floor value:
\[ \text{Number of balls} = 2546 \]

### Final Answer:
Therefore, the number of balls of radius 1 cm that can be made by melting a cube of side 22 cm is **2546**.

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