A hollow metal sphere of internal and external radii 2 cm and 4 cm respectively is melted into a solid cone of base radius 4 cm. Find the height and slant height of the cone.

Answer

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

### Step 1: Calculate the Volume of the Hollow Sphere
The volume \( V \) of a hollow sphere 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. Here, \( R = 4 \) cm and \( r = 2 \) cm.

Substituting the values:

\[ V = \frac{4}{3} \pi (4^3 - 2^3) \]

Calculating \( 4^3 \) and \( 2^3 \):

\[ 4^3 = 64 \quad \text{and} \quad 2^3 = 8 \]

So,

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

### Step 2: Set the Volume of the Cone Equal to the Volume of the Hollow Sphere
The volume \( V \) of a cone is given by the formula:

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

where \( r \) is the base radius and \( h \) is the height. Here, the base radius \( r = 4 \) cm.

Setting the volume of the cone equal to the volume of the hollow sphere:

\[ \frac{224}{3} \pi = \frac{1}{3} \pi (4^2) h \]

### Step 3: Simplify and Solve for Height \( h \)
Cancelling \( \pi \) and \( \frac{1}{3} \) from both sides:

\[ 224 = 16h \]

Now, solving for \( h \):

\[ h = \frac{224}{16} = 14 \text{ cm} \]

### Step 4: Calculate the Slant Height \( L \) of the Cone
The slant height \( L \) of a cone can be calculated using the Pythagorean theorem:

\[ L = \sqrt{r^2 + h^2} \]

Substituting the values of \( r \) and \( h \):

\[ L = \sqrt{4^2 + 14^2} = \sqrt{16 + 196} = \sqrt{212} \]

Calculating \( \sqrt{212} \):

\[ L \approx 14.56 \text{ cm} \]

### Final Answers
- Height of the cone \( h = 14 \) cm
- Slant height of the cone \( L \approx 14.56 \) cm

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