How many balls each of radius 1 cm can be made by melting a bigger ball whose diameter is 8 cm?

Answer

To solve the problem of how many balls of radius 1 cm can be made by melting a bigger ball with a diameter of 8 cm, we will follow these steps:

### Step-by-Step Solution:

1. **Determine the radius of the bigger ball:**
The diameter of the bigger ball is given as 8 cm. The radius (R) is half of the diameter.
\[ R = \frac{8 \text{ cm}}{2} = 4 \text{ cm} \]

2. **Determine the radius of the smaller balls:**
The radius of each smaller ball is given as 1 cm.
\[ r = 1 \text{ cm} \]

3. **Calculate the volume of the bigger ball:**
The volume (V) of a sphere is calculated using the formula:
\[ V = \frac{4}{3} \pi R^3 \]
Substituting the radius of the bigger ball:
\[ V_{\text{big}} = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]

4. **Calculate the volume of one smaller ball:**
Using the same volume formula for the smaller ball:
\[ V_{\text{small}} = \frac{4}{3} \pi r^3 \]
Substituting the radius of the smaller ball:
\[ V_{\text{small}} = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \text{ cm}^3 \]

5. **Set up the equation for the total volume:**
When the bigger ball is melted, its volume will equal the total volume of the smaller balls produced. If \( n \) is the number of smaller balls, we have:
\[ V_{\text{big}} = n \cdot V_{\text{small}} \]
Substituting the volumes we calculated:
\[ \frac{256}{3} \pi = n \cdot \frac{4}{3} \pi \]

6. **Cancel out \(\frac{4}{3} \pi\) from both sides:**
\[ 256 = n \cdot 4 \]

7. **Solve for \( n \):**
\[ n = \frac{256}{4} = 64 \]

### Final Answer:
Thus, **64 balls** of radius 1 cm can be made by melting the bigger ball.

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