How many solid spherical balls of radius 2cm and be made by melting a solid sphere of radius 8 cm?

Options

A. 4
B. 16
C. 32
D. 64

Answer

To solve the problem of how many solid spherical balls of radius 2 cm can be made by melting a solid sphere of radius 8 cm, we will follow these steps:

### Step 1: Calculate the Volume of the Large Sphere
The formula for the volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

For the large sphere with a radius of \( R = 8 \) cm:

\[ V_{\text{large}} = \frac{4}{3} \pi (8)^3 \]

Calculating \( (8)^3 \):

\[ (8)^3 = 512 \]

Thus,

\[ V_{\text{large}} = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \, \text{cm}^3 \]

### Step 2: Calculate the Volume of One Small Sphere
Now, we calculate the volume of one small sphere with a radius of \( r = 2 \) cm:

\[ V_{\text{small}} = \frac{4}{3} \pi (2)^3 \]

Calculating \( (2)^3 \):

\[ (2)^3 = 8 \]

Thus,

\[ V_{\text{small}} = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \, \text{cm}^3 \]

### Step 3: Determine the Number of Small Spheres
To find the number of small spheres that can be made from the large sphere, we divide the volume of the large sphere by the volume of one small sphere:

\[ \text{Number of small spheres} = \frac{V_{\text{large}}}{V_{\text{small}}} \]

Substituting the volumes we calculated:

\[ \text{Number of small spheres} = \frac{\frac{2048}{3} \pi}{\frac{32}{3} \pi} \]

The \( \pi \) and \( \frac{1}{3} \) cancel out:

\[ \text{Number of small spheres} = \frac{2048}{32} \]

Calculating \( \frac{2048}{32} \):

\[ \frac{2048}{32} = 64 \]

### Conclusion
Therefore, the number of solid spherical balls of radius 2 cm that can be made by melting a solid sphere of radius 8 cm is **64**.

---