A metallic sphere of radius 7 cm is melted and recast in to right circular cone of same radius. Find the height of the cone.

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the sphere (r) = 7 cm
- Radius of the cone (r) = 7 cm (since it is mentioned that the radius remains the same)

2. **Formulas for Volume:**
- Volume of a sphere (V_sphere) is given by the formula:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]
- Volume of a cone (V_cone) is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
where \( h \) is the height of the cone.

3. **Set the Volumes Equal:**
Since the sphere is melted and recast into the cone, the volumes will be equal:
\[ V_{sphere} = V_{cone} \]
Therefore,
\[ \frac{4}{3} \pi r^3 = \frac{1}{3} \pi r^2 h \]

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

5. **Substitute the Radius:**
Substitute \( r = 7 \) cm into the equation:
\[ 4(7^3) = (7^2) h \]

6. **Calculate \( 7^3 \) and \( 7^2 \):**
- \( 7^3 = 343 \)
- \( 7^2 = 49 \)

Now substitute these values:
\[ 4 \times 343 = 49h \]
This simplifies to:
\[ 1372 = 49h \]

7. **Solve for Height \( h \):**
Divide both sides by 49:
\[ h = \frac{1372}{49} \]
Calculating this gives:
\[ h = 28 \text{ cm} \]

### Final Answer:
The height of the cone is **28 cm**.