A metallic hemisphere is melted and recast in the shape of a cone with the ame base radius (R) as that of the hemisphere. If H is the height of the cone, then :

Options

A. a)H = 2R
B. b)`H = (2)/(3)R`
C. c)`H = sqrt(3)R`
D. d)B = 3R

Answer

To solve the problem, we need to find the height \( H \) of a cone that is formed by melting a metallic hemisphere. The base radius of the cone is the same as that of the hemisphere.

### Step-by-Step Solution:

1. **Volume of the Hemisphere**:
The volume \( V \) of a hemisphere with radius \( R \) is given by the formula:
\[ V = \frac{2}{3} \pi R^3 \]

2. **Volume of the Cone**:
The volume \( V \) of a cone with base radius \( R \) and height \( H \) is given by the formula:
\[ V = \frac{1}{3} \pi R^2 H \]

3. **Setting the Volumes Equal**:
Since the hemisphere is melted and recast into a cone, the volumes of both shapes are equal:
\[ \frac{2}{3} \pi R^3 = \frac{1}{3} \pi R^2 H \]

4. **Canceling Common Terms**:
We can cancel \( \frac{1}{3} \pi \) from both sides of the equation:
\[ 2R^3 = R^2 H \]

5. **Solving for Height \( H \)**:
To isolate \( H \), we divide both sides by \( R^2 \) (assuming \( R \neq 0 \)):
\[ H = \frac{2R^3}{R^2} = 2R \]

Thus, the height \( H \) of the cone is:
\[ H = 2R \]

### Final Answer:
The height \( H \) of the cone is \( 2R \).