A solid metallic sphere of radius 14 cm is melted and recast into a cone with diameter of the base as 14 cm. What is the height (in cm) of the cone ?

Options

A. 236
B. 64
C. 112
D. 224

Answer

To find the height of the cone formed by melting a solid metallic sphere, we can follow these steps:

### Step 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 14 cm, we can substitute this value into the formula:
\[ V = \frac{4}{3} \pi (14)^3 \]

### Step 2: Calculate \( 14^3 \).
First, we need to calculate \( 14^3 \):
\[ 14^3 = 14 \times 14 \times 14 = 196 \times 14 = 2744 \]

### Step 3: Substitute \( 14^3 \) back into the volume formula.
Now, substituting \( 14^3 \) back into the volume formula:
\[ V = \frac{4}{3} \pi (2744) = \frac{10976}{3} \pi \text{ cm}^3 \]

### Step 4: Calculate the volume of the cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

Given that the diameter of the base of the cone is 14 cm, the radius \( r \) is:
\[ r = \frac{14}{2} = 7 \text{ cm} \]

### Step 5: Substitute the radius into the cone volume formula.
Substituting \( r \) into the volume formula for the cone:
\[ V = \frac{1}{3} \pi (7)^2 h = \frac{1}{3} \pi (49) h = \frac{49}{3} \pi h \text{ cm}^3 \]

### Step 6: Set the volumes of the sphere and cone equal to each other.
Since the sphere is melted and recast into the cone, their volumes are equal:
\[ \frac{10976}{3} \pi = \frac{49}{3} \pi h \]

### Step 7: Cancel \( \pi \) and solve for \( h \).
Cancelling \( \pi \) from both sides:
\[ \frac{10976}{3} = \frac{49}{3} h \]

Multiplying both sides by 3 to eliminate the fraction:
\[ 10976 = 49h \]

### Step 8: Solve for \( h \).
Now, divide both sides by 49:
\[ h = \frac{10976}{49} \]

Calculating \( h \):
\[ h = 224 \text{ cm} \]

### Final Answer:
The height of the cone is \( 224 \) cm.

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