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

Explanation

To solve the problem, we need to find the diameter of the third spherical ball after melting a solid spherical ball of iron with a radius of 6 cm and recasting it into three smaller spherical balls with known radii of 3 cm and 4 cm.

### Step-by-Step Solution:

1. **Calculate the Volume of the Original Ball:**
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
For the original ball with radius \( r_1 = 6 \) cm:
\[ V_1 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \text{ cm}^3 \]

2. **Calculate the Volumes of the Smaller Balls:**
- For the first smaller ball with radius \( r_2 = 3 \) cm:
\[ V_2 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]
- For the second smaller ball with radius \( r_3 = 4 \) cm:
\[ V_3 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]

3. **Set Up the Volume Equation:**
Since the original volume is equal to the sum of the volumes of the three smaller balls:
\[ V_1 = V_2 + V_3 + V_4 \]
Where \( V_4 \) is the volume of the third ball with radius \( r_4 \):
\[ 288 \pi = 36 \pi + \frac{256}{3} \pi + V_4 \]

4. **Simplify the Equation:**
Dividing through by \( \pi \):
\[ 288 = 36 + \frac{256}{3} + V_4 \]
To combine the terms, convert \( 36 \) to a fraction:
\[ 36 = \frac{108}{3} \]
So,
\[ 288 = \frac{108}{3} + \frac{256}{3} + V_4 \]
Combine the fractions:
\[ 288 = \frac{364}{3} + V_4 \]
Now, isolate \( V_4 \):
\[ V_4 = 288 - \frac{364}{3} \]
Convert \( 288 \) to a fraction:
\[ 288 = \frac{864}{3} \]
Thus,
\[ V_4 = \frac{864}{3} - \frac{364}{3} = \frac{500}{3} \text{ cm}^3 \]

5. **Find the Radius of the Third Ball:**
Using the volume formula for the third ball:
\[ V_4 = \frac{4}{3} \pi r_4^3 \]
Set the volumes equal:
\[ \frac{500}{3} = \frac{4}{3} \pi r_4^3 \]
Cancel \( \frac{4}{3} \):
\[ 500 = 4 \pi r_4^3 \]
Solve for \( r_4^3 \):
\[ r_4^3 = \frac{500}{4 \pi} = \frac{125}{\pi} \]
Taking the cube root:
\[ r_4 = \sqrt[3]{\frac{125}{\pi}} \approx 5 \text{ cm} \]

6. **Calculate the Diameter of the Third Ball:**
The diameter \( d_4 \) is twice the radius:
\[ d_4 = 2 r_4 = 2 \times 5 = 10 \text{ cm} \]

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