A solid sphere of radius 3 cm is melted and recast into a cylinder of radius 2 cm. The height of the cylinder is `:`

Options

A. 8 cm
B. 9 cm
C. 7 cm
D. 10 cm

Answer

To find the height of the cylinder formed by melting a solid sphere, we will use the principle that the volume of the sphere is equal to the volume of the cylinder.

### Step-by-Step Solution:

1. **Calculate 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. Here, the radius \( r = 3 \) cm.
\[ V = \frac{4}{3} \pi (3)^3 \]
\[ V = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]

2. **Set Up 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. Here, the radius \( r = 2 \) cm and we need to find the height \( h \).

3. **Equate the Volumes:**
Since the volume of the sphere is equal to the volume of the cylinder, we can set up the equation:
\[ 36 \pi = \pi (2)^2 h \]
\[ 36 \pi = \pi (4) h \]

4. **Simplify the Equation:**
We can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ 36 = 4h \]

5. **Solve for Height \( h \):**
Now, divide both sides by 4 to find \( h \):
\[ h = \frac{36}{4} = 9 \text{ cm} \]

### Final Answer:
The height of the cylinder is \( 9 \) cm.