A cone of height 15 cm is melted and reacast in to cylinder of same base .Find the height of the cylinder.

Answer

To solve the problem of finding the height of a cylinder formed by melting a cone, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We know that a cone is melted and recast into a cylinder with the same base. We need to find the height of the cylinder.

2. **Volume Formulas**:
- The volume of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
- The volume of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 H \]
where \( r \) is the radius of the base, \( h \) is the height of the cone, and \( H \) is the height of the cylinder.

3. **Set Volumes Equal**: Since the cone is melted and recast into a cylinder, their volumes are equal:
\[ V_{\text{cone}} = V_{\text{cylinder}} \]
This gives us:
\[ \frac{1}{3} \pi r^2 h = \pi r^2 H \]

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

5. **Substitute Known Values**: We know the height of the cone \( h = 15 \) cm. Substitute this value into the equation:
\[ \frac{1}{3} \times 15 = H \]

6. **Calculate the Height of the Cylinder**:
\[ H = \frac{15}{3} = 5 \text{ cm} \]

### Final Answer:
The height of the cylinder is **5 cm**.

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