Two solid cylinders, one with diameter 60 cm and height 30 cm and the other with radius 30 cm and height 60 cm, are melted and recasted into a third solid cylinder of height 10 cm. Find the diameter of the cylinder formed.

Answer

To solve the problem, we need to find the diameter of a new cylinder formed by melting and recasting two existing cylinders. Let's break down the steps:

### Step-by-Step Solution:

1. **Identify the dimensions of the first cylinder:**
- Diameter \(d_1 = 60 \, \text{cm}\)
- Radius \(r_1 = \frac{d_1}{2} = \frac{60}{2} = 30 \, \text{cm}\)
- Height \(h_1 = 30 \, \text{cm}\)

2. **Identify the dimensions of the second cylinder:**
- Radius \(r_2 = 30 \, \text{cm}\) (given)
- Height \(h_2 = 60 \, \text{cm}\)

3. **Identify the dimensions of the third cylinder:**
- Height \(h_3 = 10 \, \text{cm}\) (given)
- Radius \(r_3\) is what we need to find.

4. **Volume of a cylinder formula:**
\[ V = \pi r^2 h \]

5. **Set up the equation for volumes:**
The total volume of the first and second cylinders equals the volume of the third cylinder:
\[ V_1 + V_2 = V_3 \]
\[ \pi r_1^2 h_1 + \pi r_2^2 h_2 = \pi r_3^2 h_3 \]

6. **Factor out \(\pi\):**
\[ r_1^2 h_1 + r_2^2 h_2 = r_3^2 h_3 \]

7. **Substitute the known values:**
\[ (30^2 \cdot 30) + (30^2 \cdot 60) = r_3^2 \cdot 10 \]

8. **Calculate the left side:**
- Calculate \(30^2 = 900\)
- Calculate \(900 \cdot 30 = 27000\)
- Calculate \(900 \cdot 60 = 54000\)
- Add them together:
\[ 27000 + 54000 = 81000 \]

9. **Set up the equation:**
\[ 81000 = r_3^2 \cdot 10 \]

10. **Solve for \(r_3^2\):**
\[ r_3^2 = \frac{81000}{10} = 8100 \]

11. **Find \(r_3\):**
\[ r_3 = \sqrt{8100} = 90 \, \text{cm} \]

12. **Calculate the diameter of the new cylinder:**
\[ d_3 = 2 \cdot r_3 = 2 \cdot 90 = 180 \, \text{cm} \]

### Final Answer:
The diameter of the cylinder formed is **180 cm**.