A metallic cone is melted and recast in to a cylinder of same radius and height 9 cm . Find the height of the cone.

Answer

To find the height of the cone when it is melted and recast into a cylinder of the same radius and height, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We know that the volume of the cone will equal the volume of the cylinder since the cone is melted and recast into the cylinder.

2. **Write the Volume Formulas**:
- The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h_1 \]
- The volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h_2 \]

3. **Set the Volumes Equal**:
Since the volumes are equal, we can write:
\[ \frac{1}{3} \pi r^2 h_1 = \pi r^2 h_2 \]

4. **Cancel Common Terms**:
We can cancel \( \pi \) and \( r^2 \) from both sides (assuming \( r \neq 0 \)):
\[ \frac{1}{3} h_1 = h_2 \]

5. **Substitute the Height of the Cylinder**:
We know from the problem that the height of the cylinder \( h_2 \) is 9 cm:
\[ \frac{1}{3} h_1 = 9 \]

6. **Solve for the Height of the Cone**:
To find \( h_1 \), multiply both sides by 3:
\[ h_1 = 9 \times 3 = 27 \text{ cm} \]

7. **Conclusion**:
The height of the cone is \( 27 \) cm.

### Final Answer:
The height of the cone is **27 cm**.