A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. Find the height of the cone.

Answer

To solve the problem of finding the height of the cone formed by melting a hollow sphere, we can follow these steps:

### Step 1: Understand the dimensions of the hollow sphere
- The internal diameter of the hollow sphere is given as 4 cm, and the external diameter is 8 cm.
- To find the internal radius (r) and external radius (R):
- Internal radius, \( r = \frac{4}{2} = 2 \) cm
- External radius, \( R = \frac{8}{2} = 4 \) cm

**Hint:** Remember that the radius is half of the diameter.

### Step 2: Calculate the volume of the hollow sphere
- The volume \( V \) of a hollow sphere can be calculated using the formula:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]
- Substitute the values of \( R \) and \( r \):
\[ V = \frac{4}{3} \pi (4^3 - 2^3) = \frac{4}{3} \pi (64 - 8) = \frac{4}{3} \pi (56) \]
- Thus, the volume of the hollow sphere is:
\[ V = \frac{224}{3} \pi \text{ cm}^3 \]

**Hint:** Make sure to cube the radius values correctly when calculating the volume.

### Step 3: Set up the volume of the cone
- The cone formed has a base diameter of 8 cm, so the radius of the cone \( r_c \) is:
\[ r_c = \frac{8}{2} = 4 \text{ cm} \]
- The volume \( V_c \) of the cone can be calculated using the formula:
\[ V_c = \frac{1}{3} \pi r_c^2 h \]
- Substitute \( r_c \) into the formula:
\[ V_c = \frac{1}{3} \pi (4^2) h = \frac{1}{3} \pi (16) h = \frac{16}{3} \pi h \]

**Hint:** Remember that the volume of the cone depends on both the radius and the height.

### Step 4: Equate the volumes of the hollow sphere and the cone
- Since the hollow sphere is melted into the cone, their volumes are equal:
\[ \frac{224}{3} \pi = \frac{16}{3} \pi h \]
- Cancel \( \frac{\pi}{3} \) from both sides:
\[ 224 = 16h \]

**Hint:** You can simplify equations by canceling out common factors.

### Step 5: Solve for the height \( h \) of the cone
- Rearranging gives:
\[ h = \frac{224}{16} = 14 \text{ cm} \]

**Hint:** When dividing, simplify the numbers to make calculations easier.

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