A metallic spherical shell of internal and external diameters 4 cm and 8 cm respectively , is melted and recast into the form of a cone of base diameter 8 cm . The height of the cone is

Options

A. `12 cm`
B. 14cm
C. 15cm
D. 8cm

Explanation

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

### Step 1: Calculate the volume of the spherical shell.
The volume of a spherical shell is given by the formula:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]
where \( R \) is the external radius and \( r \) is the internal radius.

Given:
- Internal diameter = 4 cm, so internal radius \( r = \frac{4}{2} = 2 \) cm.
- External diameter = 8 cm, so external radius \( R = \frac{8}{2} = 4 \) cm.

Now, substituting the values:
\[ V = \frac{4}{3} \pi (4^3 - 2^3) \]
\[ = \frac{4}{3} \pi (64 - 8) \]
\[ = \frac{4}{3} \pi (56) \]
\[ = \frac{224}{3} \pi \, \text{cm}^3 \]

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

Given:
- Base diameter of the cone = 8 cm, so radius \( r = \frac{8}{2} = 4 \) cm.

Substituting the radius into the volume formula:
\[ V = \frac{1}{3} \pi (4^2) h \]
\[ = \frac{1}{3} \pi (16) h \]
\[ = \frac{16}{3} \pi h \, \text{cm}^3 \]

### Step 3: Set the volumes equal to each other.
Since the volume of the spherical shell is equal to the volume of the cone, we can set the two equations equal:
\[ \frac{224}{3} \pi = \frac{16}{3} \pi h \]

### Step 4: Solve for \( h \).
We can cancel \( \pi \) and \( \frac{1}{3} \) from both sides:
\[ 224 = 16h \]

Now, divide both sides by 16:
\[ h = \frac{224}{16} \]
\[ h = 14 \, \text{cm} \]

### Final Answer:
The height of the cone is \( 14 \, \text{cm} \).

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