A sphere of radius 3 cm is drawn into a wire of thickness of `0.5 ` cm . What is the length of the wire ?

Answer

To find the length of the wire made from a sphere of radius 3 cm and thickness 0.5 cm, we will use the principle of conservation of volume. The volume of the sphere will be equal to the volume of the cylindrical wire formed.

### Step-by-Step Solution:

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

2. **Determine the radius of the wire**:
The thickness of the wire is given as 0.5 cm. Therefore, the radius of the wire (cylinder) will be:
\[ r_{\text{wire}} = \frac{0.5}{2} = 0.25 \text{ cm} \]

3. **Calculate the volume of the wire (cylinder)**:
The volume of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height (length of the wire). Substituting the radius of the wire:
\[ V = \pi (0.25)^2 h = \pi (0.0625) h = 0.0625\pi h \text{ cm}^3 \]

4. **Set the volumes equal**:
Since the volume of the sphere is equal to the volume of the wire:
\[ 36\pi = 0.0625\pi h \]

5. **Cancel \(\pi\) from both sides**:
\[ 36 = 0.0625 h \]

6. **Solve for \( h \)**:
\[ h = \frac{36}{0.0625} = 576 \text{ cm} \]

Thus, the length of the wire is **576 cm**.