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

Explanation

To solve the problem step by step, we will first calculate the volume of the hollow sphere and then equate it to the volume of the cone formed after melting the sphere.

### 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.

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

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

### Step 2: Calculate the volume of the cone

The volume \( V \) 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, thus the radius \( r = \frac{8}{2} = 4 \) cm

Now, 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 3: Equate the volumes

Since the volume of the hollow sphere is equal to the volume of the cone, we can set the two equations equal to each other:
\[ \frac{224}{3} \pi = \frac{16}{3} \pi h \]

### Step 4: Solve for height \( h \)

We can cancel \( \pi \) from both sides:
\[ \frac{224}{3} = \frac{16}{3} h \]
Multiplying both sides by 3:
\[ 224 = 16h \]
Now, divide both sides by 16:
\[ h = \frac{224}{16} = 14 \text{ cm} \]

### Step 5: Calculate the slant height \( l \)

The slant height \( l \) of the cone can be calculated using the formula:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the values of \( h \) and \( r \):
\[ l = \sqrt{14^2 + 4^2} = \sqrt{196 + 16} = \sqrt{212} \]
Calculating \( \sqrt{212} \):
\[ l = \sqrt{4 \times 53} = 2\sqrt{53} \text{ cm} \]

### Final Answers:
- Height of the cone \( h = 14 \) cm
- Slant height of the cone \( l = 2\sqrt{53} \) cm