By melting a hemisphere having radius 6.3 cm is converted into a cylinder. If the radius of the cylinder is 9 cm then find the height of the cylinder

Options

A. 1) 3.452 cm
B. 2) 4.116 cm
C. 3) 4.638 cm
D. 4) 5.476 cm

Answer

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

### Step 1: Calculate the Volume of the Hemisphere
The formula for the volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
where \( r \) is the radius of the hemisphere.

Given that the radius of the hemisphere is \( 6.3 \, \text{cm} \):
\[ V = \frac{2}{3} \pi (6.3)^3 \]

### Step 2: Calculate \( (6.3)^3 \)
Calculating \( (6.3)^3 \):
\[ (6.3)^3 = 6.3 \times 6.3 \times 6.3 = 250.047 \]
Now substituting back into the volume formula:
\[ V = \frac{2}{3} \pi (250.047) \approx \frac{500.094}{3} \pi \approx 166.698 \pi \, \text{cm}^3 \]

### Step 3: Calculate the Volume of the Cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

Given that the radius of the cylinder is \( 9 \, \text{cm} \):
\[ V = \pi (9)^2 h = 81\pi h \]

### Step 4: Set the Volumes Equal
Since the volume of the melted hemisphere is equal to the volume of the cylinder, we set the two equations equal:
\[ 166.698 \pi = 81 \pi h \]

### Step 5: Solve for \( h \)
Dividing both sides by \( \pi \):
\[ 166.698 = 81h \]
Now, solving for \( h \):
\[ h = \frac{166.698}{81} \approx 2.06 \, \text{cm} \]

### Step 6: Final Calculation
To convert this into a more precise decimal:
\[ h \approx 2.06 \, \text{cm} \]

### Conclusion
The height of the cylinder is approximately \( 2.06 \, \text{cm} \).