A spherical lead ball of radius 6 cm is melted and small lead balls of radius 3 mm are made. The total number of possible small lead balls is :

Options

A. 4250
B. 4000
C. 8005
D. 8000

Answer

To find the total number of small lead balls that can be made from a larger spherical lead ball, we need to calculate the volumes of both the large and small balls and then determine how many small balls can be formed from the volume of the large ball.

### Step-by-Step Solution:

1. **Calculate the volume of the large lead ball:**
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For the large ball, the radius \( r = 6 \) cm.
\[ V_{\text{large}} = \frac{4}{3} \pi (6)^3 \]
\[ = \frac{4}{3} \pi (216) \]
\[ = \frac{864}{3} \pi \]
\[ = 288 \pi \text{ cm}^3 \]

2. **Convert the radius of the small lead ball to centimeters:**
The radius of the small ball is given as \( 3 \) mm. To convert this to centimeters:
\[ 3 \text{ mm} = \frac{3}{10} \text{ cm} = 0.3 \text{ cm} \]

3. **Calculate the volume of the small lead ball:**
Using the same volume formula for the small ball:
\[ V_{\text{small}} = \frac{4}{3} \pi (0.3)^3 \]
\[ = \frac{4}{3} \pi (0.027) \]
\[ = \frac{0.108}{3} \pi \]
\[ = 0.036 \pi \text{ cm}^3 \]

4. **Determine the total number of small balls:**
To find the number of small balls that can be made from the volume of the large ball, we divide the volume of the large ball by the volume of one small ball:
\[ \text{Number of small balls} = \frac{V_{\text{large}}}{V_{\text{small}}} \]
\[ = \frac{288 \pi}{0.036 \pi} \]
The \( \pi \) cancels out:
\[ = \frac{288}{0.036} \]
\[ = 8000 \]

Thus, the total number of small lead balls that can be made is **8000**.