A solid sphere of radius 3 cm is melted and then cast into smaller spherical balls, each of diameter 0.6 cm. Find the number of small balls thus obtained.

Explanation

To find the number of smaller spherical balls that can be made from a solid sphere of radius 3 cm, we will follow these steps:

### Step 1: Calculate the volume of the larger sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

For the larger sphere:
- Radius \( r = 3 \) cm

Substituting the value into the formula:
\[ V = \frac{4}{3} \pi (3)^3 \]
Calculating \( (3)^3 \):
\[ (3)^3 = 27 \]
Now substituting back:
\[ V = \frac{4}{3} \pi (27) = \frac{108}{3} \pi = 36 \pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of one smaller sphere.
The diameter of the smaller sphere is 0.6 cm, so the radius \( r \) of the smaller sphere is:
\[ r = \frac{0.6}{2} = 0.3 \, \text{cm} \]

Using the volume formula again for the smaller sphere:
\[ V = \frac{4}{3} \pi (0.3)^3 \]
Calculating \( (0.3)^3 \):
\[ (0.3)^3 = 0.027 \]
Now substituting back:
\[ V = \frac{4}{3} \pi (0.027) = \frac{0.108}{3} \pi = 0.036 \pi \, \text{cm}^3 \]

### Step 3: Find the number of smaller spheres.
Let \( n \) be the number of smaller spheres. The total volume of the smaller spheres must equal the volume of the larger sphere:
\[ n \times \text{Volume of one smaller sphere} = \text{Volume of larger sphere} \]
Substituting the volumes:
\[ n \times 0.036 \pi = 36 \pi \]

### Step 4: Solve for \( n \).
Dividing both sides by \( \pi \):
\[ n \times 0.036 = 36 \]
Now, divide both sides by \( 0.036 \):
\[ n = \frac{36}{0.036} \]
Calculating the right-hand side:
\[ n = 1000 \]

### Conclusion:
Thus, the number of smaller balls obtained is \( n = 1000 \).

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