The number of balls each of radius 2 cm can be made by melting a big ball whose radius Is 8 cm Is equal to

Options

A. 60
B. 64
C. 70
D. 74

Answer

To solve the problem of how many small balls of radius 2 cm can be made from a big ball of radius 8 cm, we will follow these steps:

### Step 1: Calculate the volume of the big ball
The formula for the volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

For the big ball, the radius \( r \) is 8 cm. Therefore, we can substitute this value into the formula:

\[ V_{\text{big}} = \frac{4}{3} \pi (8)^3 \]

### Step 2: Calculate \( (8)^3 \)
Calculating \( (8)^3 \):

\[ (8)^3 = 512 \]

### Step 3: Substitute back into the volume formula
Now substituting back into the volume formula:

\[ V_{\text{big}} = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \text{ cm}^3 \]

### Step 4: Calculate the volume of one small ball
Now, we will calculate the volume of one small ball with a radius of 2 cm using the same volume formula:

\[ V_{\text{small}} = \frac{4}{3} \pi (2)^3 \]

### Step 5: Calculate \( (2)^3 \)
Calculating \( (2)^3 \):

\[ (2)^3 = 8 \]

### Step 6: Substitute back into the volume formula for the small ball
Now substituting back into the volume formula:

\[ V_{\text{small}} = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \text{ cm}^3 \]

### Step 7: Calculate the number of small balls
To find the number of small balls that can be made from the big ball, we divide the volume of the big ball by the volume of one small ball:

\[ \text{Number of small balls} = \frac{V_{\text{big}}}{V_{\text{small}}} = \frac{\frac{2048}{3} \pi}{\frac{32}{3} \pi} \]

### Step 8: Simplify the expression
The \( \frac{3}{3} \) and \( \pi \) cancel out:

\[ \text{Number of small balls} = \frac{2048}{32} \]

### Step 9: Perform the division
Now we perform the division:

\[ \frac{2048}{32} = 64 \]

### Conclusion
Therefore, the number of small balls that can be made by melting the big ball is **64**.

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