A metallic solid sphere of radius 9 cm is melted to form a solid cylinder of radius 9 cm . The height of the cylinder is

Options

A. 12 cm
B. 18 cm
C. 36 cm
D. 96 cm

Explanation

To find the height of the cylinder formed by melting a metallic solid sphere, we need to equate the volumes of both shapes. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Find the Volume of the Sphere:**
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Given that the radius of the sphere is 9 cm, we substitute \( r = 9 \) cm into the formula:
\[ V = \frac{4}{3} \pi (9)^3 \]

2. **Calculate \( 9^3 \):**
First, we calculate \( 9^3 \):
\[ 9^3 = 729 \]
So, the volume of the sphere becomes:
\[ V = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi = 972 \pi \, \text{cm}^3 \]

3. **Find 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. The radius of the cylinder is also 9 cm, so we substitute \( r = 9 \) cm into the formula:
\[ V = \pi (9)^2 h \]

4. **Calculate \( 9^2 \):**
Now, we calculate \( 9^2 \):
\[ 9^2 = 81 \]
Therefore, the volume of the cylinder can be expressed as:
\[ V = \pi (81) h = 81 \pi h \, \text{cm}^3 \]

5. **Equate the Volumes:**
Since the sphere is melted to form the cylinder, their volumes are equal:
\[ 972 \pi = 81 \pi h \]

6. **Cancel \( \pi \) from both sides:**
We can divide both sides by \( \pi \):
\[ 972 = 81 h \]

7. **Solve for \( h \):**
To find \( h \), divide both sides by 81:
\[ h = \frac{972}{81} \]

8. **Calculate \( h \):**
Performing the division:
\[ h = 12 \, \text{cm} \]

### Final Answer:
The height of the cylinder is \( 12 \, \text{cm} \).