12 sphere of the same size are made by melting a solid cylidner of 16 cm diameter and 2 cm height. The diameter of each sphere is :

Options

A. 2 cm
B. 4 cm
C. 3 cm
D. `sqrt(3) cm`

Answer

To solve the problem of finding the diameter of each sphere made from melting a solid cylinder, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions of the cylinder**:
- The diameter of the cylinder is given as 16 cm.
- The height of the cylinder is given as 2 cm.

2. **Calculate the radius of the cylinder**:
- The radius (r) of the cylinder can be calculated using the formula:
\[ r = \frac{\text{diameter}}{2} = \frac{16 \text{ cm}}{2} = 8 \text{ cm} \]

3. **Calculate the volume of the cylinder**:
- The volume (V) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
- Substituting the values we have:
\[ V = \pi (8 \text{ cm})^2 (2 \text{ cm}) = \pi \times 64 \text{ cm}^2 \times 2 \text{ cm} = 128\pi \text{ cm}^3 \]

4. **Determine the volume of one sphere**:
- Since 12 spheres are made from the melted cylinder, the total volume of the spheres will equal the volume of the cylinder:
\[ \text{Total volume of spheres} = 128\pi \text{ cm}^3 \]
- Therefore, the volume of one sphere (V_s) is:
\[ V_s = \frac{128\pi \text{ cm}^3}{12} = \frac{32\pi \text{ cm}^3}{3} \]

5. **Use the volume formula for a sphere to find its radius**:
- The volume of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]
- Setting the volume of one sphere equal to the formula:
\[ \frac{4}{3}\pi r^3 = \frac{32\pi}{3} \]
- Canceling \(\pi\) from both sides:
\[ \frac{4}{3} r^3 = \frac{32}{3} \]
- Multiplying both sides by \(\frac{3}{4}\):
\[ r^3 = 8 \]
- Taking the cube root:
\[ r = 2 \text{ cm} \]

6. **Calculate the diameter of each sphere**:
- The diameter (D) of each sphere is given by:
\[ D = 2r = 2 \times 2 \text{ cm} = 4 \text{ cm} \]

### Final Answer:
The diameter of each sphere is **4 cm**.