A hollow sphere of internal and external radii 6 cm and 8 cm respectively is melted and recast into small cones of base radius 2 cm and height 8 cm. Find the number of cones.

Answer

To solve the problem step by step, we will calculate the volume of the hollow sphere and the volume of one cone, and then find the number of cones that can be formed from the melted hollow sphere.

### Step 1: Calculate the Volume of the Hollow Sphere

The volume \( V_s \) of a hollow sphere is given by the formula:
\[ V_s = \frac{4}{3} \pi (R^3 - r^3) \]
where \( R \) is the external radius and \( r \) is the internal radius.

Given:
- Internal radius \( r = 6 \) cm
- External radius \( R = 8 \) cm

Substituting the values:
\[ V_s = \frac{4}{3} \pi (8^3 - 6^3) \]

Calculating \( 8^3 \) and \( 6^3 \):
\[ 8^3 = 512, \quad 6^3 = 216 \]
Now substituting these values:
\[ V_s = \frac{4}{3} \pi (512 - 216) = \frac{4}{3} \pi (296) \]

### Step 2: Calculate the Volume of One Cone

The volume \( V_c \) of a cone is given by the formula:
\[ V_c = \frac{1}{3} \pi r_c^2 h_c \]
where \( r_c \) is the base radius and \( h_c \) is the height.

Given:
- Base radius \( r_c = 2 \) cm
- Height \( h_c = 8 \) cm

Substituting the values:
\[ V_c = \frac{1}{3} \pi (2^2) (8) \]

Calculating \( 2^2 \):
\[ 2^2 = 4 \]
Now substituting this value:
\[ V_c = \frac{1}{3} \pi (4)(8) = \frac{32}{3} \pi \]

### Step 3: Find the Number of Cones

Let \( n \) be the number of cones formed. Since the volume of the hollow sphere is equal to the total volume of the cones, we have:
\[ V_s = n \cdot V_c \]
Substituting the volumes we calculated:
\[ \frac{4}{3} \pi (296) = n \cdot \frac{32}{3} \pi \]

Dividing both sides by \( \pi \) and \( \frac{1}{3} \):
\[ 4 \cdot 296 = n \cdot 32 \]

Calculating \( 4 \cdot 296 \):
\[ 4 \cdot 296 = 1184 \]
Now we have:
\[ 1184 = n \cdot 32 \]

Dividing both sides by 32 to find \( n \):
\[ n = \frac{1184}{32} \]

Calculating \( \frac{1184}{32} \):
\[ n = 37 \]

### Final Answer:
The number of cones formed is \( \boxed{37} \).

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