(i) The radius of a metallic sphere is 3 cm .It melted and recast in to wire of diameter 0.4 cm Find the length of wire.
(ii) An iron ball oif radius 4 cm is melted .How many small spheres of radus 2 cm can be formed from the material?

Answer

### Step-by-Step Solution

#### Part (i)
1. **Identify the given values**:
- Radius of the metallic sphere, \( R = 3 \) cm.
- Diameter of the wire, \( d = 0.4 \) cm. Therefore, the radius of the wire, \( r = \frac{d}{2} = \frac{0.4}{2} = 0.2 \) cm.

2. **Calculate the volume of the sphere**:
- The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
- Substituting the value of \( R \):
\[ V = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]

3. **Calculate the volume of the wire (cylinder)**:
- The volume of a cylinder is given by:
\[ V = \pi r^2 h \]
- Here, \( h \) is the length of the wire, which we need to find. Setting the volume of the sphere equal to the volume of the wire:
\[ 36 \pi = \pi (0.2)^2 h \]

4. **Simplify and solve for \( h \)**:
- Cancel \( \pi \) from both sides:
\[ 36 = (0.2)^2 h \]
- Calculate \( (0.2)^2 = 0.04 \):
\[ 36 = 0.04 h \]
- Now, solve for \( h \):
\[ h = \frac{36}{0.04} = 900 \text{ cm} \]

5. **Convert the length to meters**:
- Since \( 1 \text{ m} = 100 \text{ cm} \):
\[ h = \frac{900}{100} = 9 \text{ m} \]

#### Final Answer for Part (i):
The length of the wire is **9 meters**.

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#### Part (ii)
1. **Identify the given values**:
- Radius of the iron ball, \( R = 4 \) cm.
- Radius of the small spheres, \( r = 2 \) cm.

2. **Calculate the volume of the iron ball**:
- Using the volume formula for a sphere:
\[ V_1 = \frac{4}{3} \pi R^3 \]
- Substituting the value of \( R \):
\[ V_1 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]

3. **Calculate the volume of one small sphere**:
- Using the same volume formula:
\[ V_2 = \frac{4}{3} \pi r^3 \]
- Substituting the value of \( r \):
\[ V_2 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \text{ cm}^3 \]

4. **Calculate the number of small spheres**:
- The number of small spheres that can be formed is given by:
\[ \text{Number of spheres} = \frac{V_1}{V_2} \]
- Substituting the volumes:
\[ \text{Number of spheres} = \frac{\frac{256}{3} \pi}{\frac{32}{3} \pi} \]
- The \( \pi \) and \( \frac{1}{3} \) cancel out:
\[ \text{Number of spheres} = \frac{256}{32} = 8 \]

#### Final Answer for Part (ii):
The number of small spheres that can be formed is **8**.

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