A solid metallic sphere of radius 21 cm is melted and recast into a cone with diameter of the base as 21 cm. What is the height (in cm) of the cone ?

Options

A. 336
B. 112
C. 224
D. 66

Answer

To solve the problem, we need to find the height of a cone formed by melting a solid metallic sphere. We will use the formula for the volume of both the sphere and the cone and set them equal to each other, since the volume of the sphere will be equal to the volume of the cone after melting.

### Step-by-Step Solution:

1. **Calculate the Volume of the Sphere:**
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Here, the radius \( r \) of the sphere is 21 cm.
\[ V = \frac{4}{3} \pi (21)^3 \]

2. **Calculate \( 21^3 \):**
First, calculate \( 21^3 \):
\[ 21^3 = 21 \times 21 \times 21 = 9261 \]
Now substitute this back into the volume formula:
\[ V = \frac{4}{3} \pi (9261) = \frac{37044}{3} \pi \]

3. **Calculate the Volume of the Cone:**
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
The diameter of the cone is 21 cm, so the radius \( r \) is:
\[ r = \frac{21}{2} = 10.5 \text{ cm} \]
Now substitute the radius into the volume formula:
\[ V = \frac{1}{3} \pi (10.5)^2 h \]

4. **Calculate \( (10.5)^2 \):**
Calculate \( (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]
Substitute this back into the volume formula for the cone:
\[ V = \frac{1}{3} \pi (110.25) h = \frac{110.25}{3} \pi h \]

5. **Set the Volumes Equal:**
Since the volume of the sphere is equal to the volume of the cone, we can set them equal to each other:
\[ \frac{37044}{3} \pi = \frac{110.25}{3} \pi h \]

6. **Cancel \( \pi \) and \( \frac{1}{3} \):**
Cancel \( \pi \) and \( \frac{1}{3} \) from both sides:
\[ 37044 = 110.25 h \]

7. **Solve for \( h \):**
Now, solve for \( h \):
\[ h = \frac{37044}{110.25} \]
Calculate \( h \):
\[ h \approx 336 \text{ cm} \]

### Final Answer:
The height of the cone is approximately **336 cm**.