A solid sphere of radius 9 cm is melted to form a sphere of radius 6 cm and a right circular cylinder of same radius. The height of the cylinder so formed is

Options

A. 19 cm
B. 21 cm
C. 23 cm
D. 25 cm

Answer

To find the height of the cylinder formed when a solid sphere of radius 9 cm is melted to create a new sphere of radius 6 cm and a right circular cylinder of the same radius, we will use the formula for the volume of a sphere and a cylinder.

### Step-by-Step Solution:

1. **Calculate the Volume of the Original Sphere:**
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For the original sphere with radius \( r = 9 \) cm:
\[ V_1 = \frac{4}{3} \pi (9)^3 = \frac{4}{3} \pi (729) = 972 \pi \, \text{cm}^3 \]

2. **Calculate the Volume of the New Sphere:**
For the new sphere with radius \( r = 6 \) cm:
\[ V_2 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \, \text{cm}^3 \]

3. **Set Up the Equation for the Volume of the Cylinder:**
The total volume of the material used (which is conserved) is equal to the volume of the original sphere. Therefore, the volume of the cylinder \( V_c \) can be calculated as:
\[ V_c = V_1 - V_2 \]
Substituting the volumes we calculated:
\[ V_c = 972 \pi - 288 \pi = 684 \pi \, \text{cm}^3 \]

4. **Calculate the Volume of the Cylinder:**
The volume \( V_c \) of a cylinder is given by:
\[ V_c = \pi r^2 h \]
Here, the radius \( r = 6 \) cm, so:
\[ V_c = \pi (6)^2 h = 36 \pi h \]

5. **Equate the Volumes and Solve for Height \( h \):**
Now we set the volume of the cylinder equal to the volume we calculated:
\[ 36 \pi h = 684 \pi \]
Dividing both sides by \( \pi \):
\[ 36 h = 684 \]
Now, divide by 36:
\[ h = \frac{684}{36} = 19 \, \text{cm} \]

### Final Answer:
The height of the cylinder is \( h = 19 \, \text{cm} \).