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

Options

A. 12 cm
B. 14 cm
C. 15 cm
D. 18 cm

Explanation

To solve the problem step by step, we need to find the height of a cone formed by melting a metallic spherical shell with given internal and external diameters.

### Step 1: Identify the given dimensions
- Internal diameter of the spherical shell = 4 cm
- External diameter of the spherical shell = 8 cm

### Step 2: Calculate the radii of the spherical shell
- Internal radius (r1) = Internal diameter / 2 = 4 cm / 2 = 2 cm
- External radius (r2) = External diameter / 2 = 8 cm / 2 = 4 cm

### Step 3: Calculate the volume of the spherical shell
The volume of a spherical shell is given by the formula:
\[ V = \frac{4}{3} \pi (r_2^3 - r_1^3) \]
Substituting the values:
\[ V = \frac{4}{3} \pi (4^3 - 2^3) \]
Calculating the cubes:
- \(4^3 = 64\)
- \(2^3 = 8\)

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

### Step 4: 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 = radius of the base of the cone = 8 cm / 2 = 4 cm
- h = height of the cone (which we need to find)

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

### Step 5: 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 equations equal:
\[ \frac{224}{3} \pi = \frac{16}{3} \pi h \]

### Step 6: Cancel out \(\pi\) and solve for h
Dividing both sides by \(\pi\):
\[ \frac{224}{3} = \frac{16}{3} h \]

Now, multiply both sides by 3 to eliminate the fraction:
\[ 224 = 16h \]

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

### Final Answer
The height of the cone is **14 cm**.