A hollow metallic sphere with external diameter 8 cm and internal diameter 4 cm is melted and moulded into a cone having base radius 4 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 the cone formed by melting a hollow metallic sphere.

### Step 1: Calculate the volume of the hollow metallic sphere.

The volume of a hollow sphere can be calculated by finding the volume of the outer sphere and subtracting the volume of the inner sphere.

1. **External diameter of the sphere** = 8 cm
Therefore, the external radius \( R = \frac{8}{2} = 4 \) cm.

2. **Internal diameter of the sphere** = 4 cm
Therefore, the internal radius \( r = \frac{4}{2} = 2 \) cm.

3. **Volume of the outer sphere**:
\[ V_{\text{outer}} = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \, \text{cm}^3 \]

4. **Volume of the inner sphere**:
\[ V_{\text{inner}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \, \text{cm}^3 \]

5. **Volume of the hollow sphere**:
\[ V_{\text{hollow}} = V_{\text{outer}} - V_{\text{inner}} = \frac{256}{3} \pi - \frac{32}{3} \pi = \frac{224}{3} \pi \, \text{cm}^3 \]

### Step 2: Set the volume of the cone equal to the volume of the hollow sphere.

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

1. **Given**: Base radius of the cone \( r = 4 \) cm.

2. **Volume of the cone**:
\[ V_{\text{cone}} = \frac{1}{3} \pi (4)^2 h = \frac{1}{3} \pi (16) h = \frac{16}{3} \pi h \, \text{cm}^3 \]

### Step 3: Equate the volumes and solve for height \( h \).

Set the volume of the cone equal to the volume of the hollow sphere:
\[ \frac{16}{3} \pi h = \frac{224}{3} \pi \]

1. **Cancel \( \pi \) and \( \frac{1}{3} \)** from both sides:
\[ 16h = 224 \]

2. **Solve for \( h \)**:
\[ h = \frac{224}{16} = 14 \, \text{cm} \]

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

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