A metallic sphere of radius 12 cm is melted and cast into a cone whose base radius is 16 cm . What is the height of the cone ?

Options

A. 27 cm
B. 18 cm
C. 90 cm
D. 270 cm

Answer

To find the height of the cone formed by melting a metallic sphere, we will use the formula for the volume of a sphere and the volume of a cone.

### 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 of the sphere is 12 cm.
\[ V = \frac{4}{3} \pi (12)^3 \]
\[ = \frac{4}{3} \pi (1728) \]
\[ = \frac{6912}{3} \pi = 2304 \pi \, \text{cm}^3 \]

2. **Set Up the Volume of the Cone:**
The volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone. Here, the radius of the cone is 16 cm.
\[ V = \frac{1}{3} \pi (16)^2 h \]
\[ = \frac{1}{3} \pi (256) h \]
\[ = \frac{256}{3} \pi h \, \text{cm}^3 \]

3. **Equate the Volumes:**
Since the volume of the sphere is equal to the volume of the cone (because the sphere is melted to form the cone), we have:
\[ 2304 \pi = \frac{256}{3} \pi h \]

4. **Cancel \( \pi \) from both sides:**
\[ 2304 = \frac{256}{3} h \]

5. **Solve for \( h \):**
Multiply both sides by 3 to eliminate the fraction:
\[ 6912 = 256h \]
Now, divide both sides by 256:
\[ h = \frac{6912}{256} \]
\[ h = 27 \, \text{cm} \]

### Final Answer:
The height of the cone is **27 cm**.