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 of a sphere is given by:

\[ V_{sphere} = \frac{4}{3} \pi r^3 \]

Here, the radius \( r \) of the sphere is 14 cm.

Substituting the value:

\[ V_{sphere} = \frac{4}{3} \pi (14)^3 \]

Calculating \( 14^3 \):

\[ 14^3 = 14 \times 14 \times 14 = 2744 \]

Now substituting this back into the volume formula:

\[ V_{sphere} = \frac{4}{3} \pi (2744) = \frac{10976}{3} \pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of the cone
The formula for the volume of a cone is given by:

\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

The diameter of the base of the cone is 14 cm, so the radius \( r \) is:

\[ r = \frac{14}{2} = 7 \, \text{cm} \]

Substituting the radius into the volume formula:

\[ V_{cone} = \frac{1}{3} \pi (7)^2 h = \frac{1}{3} \pi (49) h = \frac{49}{3} \pi h \, \text{cm}^3 \]

### Step 3: Set the volumes equal to each other
Since the sphere is melted and recast into the cone, their volumes are equal:

\[ V_{sphere} = V_{cone} \]

Thus,

\[ \frac{10976}{3} \pi = \frac{49}{3} \pi h \]

### Step 4: Cancel out \(\pi\) and \(\frac{1}{3}\)
We can cancel \(\pi\) and \(\frac{1}{3}\) from both sides:

\[ 10976 = 49h \]

### Step 5: Solve for \(h\)
Now, we can solve for \(h\):

\[ h = \frac{10976}{49} \]

Calculating \(h\):

\[ h = 224 \, \text{cm} \]

### Final Answer
The height of the cone is \(224 \, \text{cm}\).

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