A solid spherical ball of iron with radius 6 cm is melted and recast into three solid spherical balls. The radii of the two balls are 3 cm and 4 cm respectively, determine the diameter of the third ball.

Answer

To solve the problem step by step, we will follow the process of calculating the volumes of the spheres involved and using the relationship between them.

### Step-by-Step Solution:

1. **Identify the radius of the original sphere:**
- The radius of the original solid spherical ball of iron is given as \( R = 6 \) cm.

2. **Identify the radii of the two smaller spheres:**
- The radius of the first smaller sphere is \( r_1 = 3 \) cm.
- The radius of the second smaller sphere is \( r_2 = 4 \) cm.

3. **Let the radius of the third sphere be \( r_3 \):**
- We need to find \( r_3 \).

4. **Use the formula for the volume of a sphere:**
- The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
- Therefore, the volume of the original sphere is:
\[ V_{\text{original}} = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (6)^3 \]

5. **Calculate the volume of the original sphere:**
- Calculate \( 6^3 = 216 \):
\[ V_{\text{original}} = \frac{4}{3} \pi (216) = 288 \pi \text{ cm}^3 \]

6. **Calculate the volumes of the two smaller spheres:**
- Volume of the first sphere:
\[ V_1 = \frac{4}{3} \pi (r_1)^3 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]
- Volume of the second sphere:
\[ V_2 = \frac{4}{3} \pi (r_2)^3 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]

7. **Set up the equation for the total volume:**
- The total volume of the three spheres must equal the volume of the original sphere:
\[ V_{\text{original}} = V_1 + V_2 + V_3 \]
- Therefore:
\[ 288 \pi = 36 \pi + \frac{256}{3} \pi + \frac{4}{3} \pi (r_3)^3 \]

8. **Simplify the equation:**
- Remove \( \pi \) from both sides:
\[ 288 = 36 + \frac{256}{3} + \frac{4}{3} (r_3)^3 \]
- Convert 36 into thirds:
\[ 288 = \frac{108}{3} + \frac{256}{3} + \frac{4}{3} (r_3)^3 \]
- Combine the fractions:
\[ 288 = \frac{364}{3} + \frac{4}{3} (r_3)^3 \]

9. **Isolate \( r_3^3 \):**
- Multiply through by 3 to eliminate the fraction:
\[ 864 = 364 + 4 (r_3)^3 \]
- Subtract 364 from both sides:
\[ 500 = 4 (r_3)^3 \]
- Divide by 4:
\[ (r_3)^3 = 125 \]

10. **Find \( r_3 \):**
- Take the cube root:
\[ r_3 = 5 \text{ cm} \]

11. **Calculate the diameter of the third sphere:**
- The diameter \( D \) is given by:
\[ D = 2r_3 = 2 \times 5 \text{ cm} = 10 \text{ cm} \]

### Final Answer:
The diameter of the third ball is **10 cm**.