A steel sphere of radius 6 cm is drawn into a wire of diameter 16 cm. find the length of the wire. (in cm)

Options

A. 3,5
B. 6
C. 4.5
D. 2.5

Answer

To find the length of the wire formed from a steel sphere, we will use the principle of conservation of volume. The volume of the steel sphere will be equal to the volume of the cylindrical wire formed.

### Step-by-Step Solution:

1. **Find the volume of the steel sphere**:
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Here, the radius \( r = 6 \) cm.
\[ V = \frac{4}{3} \pi (6)^3 \]
\[ V = \frac{4}{3} \pi (216) \]
\[ V = \frac{864}{3} \pi \]
\[ V = 288 \pi \text{ cm}^3 \]

2. **Find the volume of the cylindrical wire**:
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base of the cylinder and \( h \) is the height (or length) of the cylinder. The diameter of the wire is given as 16 cm, so the radius \( r \) is:
\[ r = \frac{16}{2} = 8 \text{ cm} \]
Let the length of the wire be \( h \). Therefore, the volume of the cylinder becomes:
\[ V = \pi (8)^2 h \]
\[ V = \pi (64) h \]
\[ V = 64 \pi h \text{ cm}^3 \]

3. **Set the volumes equal to each other**:
Since the volume of the sphere is equal to the volume of the cylinder:
\[ 288 \pi = 64 \pi h \]

4. **Solve for \( h \)**:
We can cancel \( \pi \) from both sides:
\[ 288 = 64 h \]
Now, divide both sides by 64:
\[ h = \frac{288}{64} \]
\[ h = 4.5 \text{ cm} \]

### Final Answer:
The length of the wire is **4.5 cm**.