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

Options

A. H=2R
B. H`=(2)/(3)R`
C. H=`sqrt(3R)`
D. B=3R

Answer

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

### Step-by-Step Solution:

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

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

3. **Setting the Volumes Equal**:
Since the hemisphere is melted and reshaped into a cone, their volumes 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:
\[ 2 R^3 = R^2 H \]

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

Thus, the height \( H \) of the cone is \( 2R \).

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

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