The metallic sphere of radius 6 cm is melted and recast into the sphere of a cylinder of radius 10cm . The height of the cylinder is `:`

Options

A. 2.8 cm
B. 3.2 cm
C. 4.2 cm
D. 5.7 cm

Answer

To find the height of the cylinder formed by melting a metallic sphere, we will use the principle that the volume of the sphere is equal to the volume of the cylinder.

### Step-by-Step Solution:

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

2. **Calculate \( (6)^3 \)**:
\[ (6)^3 = 216 \]
Therefore,
\[ V_{\text{sphere}} = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \, \text{cm}^3 \]

3. **Calculate the volume of the cylinder**:
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height. Here, the radius \( r = 10 \) cm.
\[ V_{\text{cylinder}} = \pi (10)^2 h = 100 \pi h \]

4. **Set the volumes equal to each other**:
Since the volume of the sphere equals the volume of the cylinder:
\[ 288 \pi = 100 \pi h \]

5. **Cancel \( \pi \) from both sides**:
\[ 288 = 100 h \]

6. **Solve for \( h \)**:
\[ h = \frac{288}{100} = 2.88 \, \text{cm} \]

### Final Answer:
The height of the cylinder is \( 2.88 \, \text{cm} \).