A metallic sphere of radius 5.6 cm is melted and recast into a shape of cylinder of radius 6cm. Find the height of the cylinder.

Answer

To solve the problem of finding the height of a cylinder formed by melting a metallic sphere, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a metallic sphere with a radius of 5.6 cm that is melted and recast into a cylinder with a radius of 6 cm. We need to find the height of the cylinder.

2. **Volume Equivalence**: Since the sphere is melted and recast into a cylinder, the volume of the sphere will be equal to the volume of the cylinder.
\[ V_{\text{sphere}} = V_{\text{cylinder}} \]

3. **Formulas for Volume**:
- The volume of a sphere is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
- The volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]

4. **Substituting the Values**:
- For the sphere, the radius \( r = 5.6 \) cm:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (5.6)^3 \]
- For the cylinder, the radius \( r = 6 \) cm and height \( h \) is unknown:
\[ V_{\text{cylinder}} = \pi (6)^2 h \]

5. **Setting the Volumes Equal**:
\[ \frac{4}{3} \pi (5.6)^3 = \pi (6)^2 h \]

6. **Canceling \( \pi \)**:
- Since \( \pi \) is present on both sides, we can cancel it out:
\[ \frac{4}{3} (5.6)^3 = (6)^2 h \]

7. **Calculating \( (5.6)^3 \)**:
- First, calculate \( 5.6^3 \):
\[ 5.6^3 = 5.6 \times 5.6 \times 5.6 = 175.616 \]
- Now substitute this value:
\[ \frac{4}{3} \times 175.616 = 36 h \]

8. **Calculating the Left Side**:
\[ \frac{4 \times 175.616}{3} = \frac{702.464}{3} \approx 234.15467 \]

9. **Setting Up the Equation**:
\[ 234.15467 = 36h \]

10. **Solving for \( h \)**:
\[ h = \frac{234.15467}{36} \approx 6.504 \]

### Final Answer:
The height of the cylinder is approximately \( 6.504 \) cm.