How many metallic balls of radius 1 cm can be recast by melting a metallic sphere of radius 8 cm?

Explanation

To solve the problem of how many metallic balls of radius 1 cm can be recast from a metallic sphere of radius 8 cm, we will follow these steps:

### Step 1: Identify the given values
- Radius of the metallic sphere (R) = 8 cm
- Radius of the metallic ball (r) = 1 cm

### Step 2: Write the formula for the volume of a sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 3: Calculate the volume of the metallic sphere
Using the radius of the metallic sphere (R = 8 cm):
\[ V_{\text{sphere}} = \frac{4}{3} \pi (8)^3 \]
Calculating \( 8^3 \):
\[ 8^3 = 512 \]
Thus, the volume of the metallic sphere is:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \text{ cm}^3 \]

### Step 4: Calculate the volume of one metallic ball
Using the radius of the metallic ball (r = 1 cm):
\[ V_{\text{ball}} = \frac{4}{3} \pi (1)^3 \]
Calculating \( 1^3 \):
\[ 1^3 = 1 \]
Thus, the volume of one metallic ball is:
\[ V_{\text{ball}} = \frac{4}{3} \pi (1) = \frac{4}{3} \pi \text{ cm}^3 \]

### Step 5: Find the number of metallic balls
To find the number of metallic balls that can be formed, we divide the volume of the metallic sphere by the volume of one metallic ball:
\[ \text{Number of balls} = \frac{V_{\text{sphere}}}{V_{\text{ball}}} = \frac{\frac{2048}{3} \pi}{\frac{4}{3} \pi} \]
The \( \pi \) and \( \frac{1}{3} \) cancel out:
\[ \text{Number of balls} = \frac{2048}{4} = 512 \]

### Final Answer
The number of metallic balls of radius 1 cm that can be recast from the metallic sphere of radius 8 cm is **512**.

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