(a) How many balls of radius 1 cm can be drawn by melting metallic sphere of radius 3 cm?
(b) The small spherical balls of diameter 0.6 cm are drawn by melting solid metallic sphere of 3cm radius .Find the number of small balls constructed.

Answer

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (a): Finding the number of balls of radius 1 cm from a metallic sphere of radius 3 cm

1. **Calculate the volume of the metallic sphere.**
- The formula for the volume \( V \) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
- Here, the radius \( r \) of the metallic sphere is 3 cm.
- Substituting the value into the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi \text{ cm}^3 \]

2. **Calculate the volume of one small ball of radius 1 cm.**
- Using the same volume formula for a sphere:
\[ V_{\text{ball}} = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi (1) = \frac{4}{3} \pi \text{ cm}^3 \]

3. **Determine the number of small balls that can be made.**
- The number of balls is given by the ratio of the volume of the metallic sphere to the volume of one small ball:
\[ \text{Number of balls} = \frac{V_{\text{sphere}}}{V_{\text{ball}}} = \frac{36\pi}{\frac{4}{3}\pi} \]
- Simplifying this:
\[ = \frac{36}{\frac{4}{3}} = 36 \times \frac{3}{4} = 27 \]

### Part (b): Finding the number of balls of diameter 0.6 cm from a metallic sphere of radius 3 cm

1. **Calculate the radius of the small ball.**
- The diameter of the small ball is 0.6 cm, so the radius \( r \) is:
\[ r = \frac{0.6}{2} = 0.3 \text{ cm} \]

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

3. **Determine the number of small balls that can be made.**
- Using the volume of the metallic sphere calculated earlier:
\[ \text{Number of small balls} = \frac{V_{\text{sphere}}}{V_{\text{small ball}}} = \frac{36\pi}{0.036\pi} \]
- Simplifying this:
\[ = \frac{36}{0.036} = 1000 \]

### Final Answers:
- (a) The number of balls of radius 1 cm is **27**.
- (b) The number of balls of diameter 0.6 cm is **1000**.