A solid sphere of radius 3 cm is melted and recast into a cylinder of radius 2cm. 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. Given that the radius of the sphere is 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 of the cylinder and \( h \) is the height. Given that the radius of the cylinder is 2 cm:
\[ V = \pi (2)^2 h = 4 \pi h \text{ cm}^3 \]

3. **Equate the volumes of the sphere and the cylinder:**
Since the sphere is melted and recast into the cylinder, their volumes are equal:
\[ 36 \pi = 4 \pi h \]

4. **Cancel \( \pi \) from both sides:**
\[ 36 = 4h \]

5. **Solve for \( h \):**
Divide both sides by 4:
\[ h = \frac{36}{4} = 9 \text{ cm} \]

### Final Answer:
The height of the cylinder is **9 cm**.

---