Three metallic spherical balls of radii 3 cm, 4 cm and 5 cm are melted and recast into a big spherical ball. Find the radius of this big ball.

Answer

To find the radius of the big spherical ball formed by melting three smaller spherical balls with radii 3 cm, 4 cm, and 5 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume of a Sphere**:
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

2. **Calculate the Volume of Each Small Ball**:
- For the first ball with radius \( r_1 = 3 \) cm:
\[ V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \, \text{cm}^3 \]
- For the second ball with radius \( r_2 = 4 \) cm:
\[ V_2 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \, \text{cm}^3 \]
- For the third ball with radius \( r_3 = 5 \) cm:
\[ V_3 = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \, \text{cm}^3 \]

3. **Find the Total Volume of the Small Balls**:
Now, we add the volumes of the three balls:
\[ V_{\text{total}} = V_1 + V_2 + V_3 = 36 \pi + \frac{256}{3} \pi + \frac{500}{3} \pi \]
To add these, we convert \( 36 \pi \) into a fraction with a denominator of 3:
\[ 36 \pi = \frac{108}{3} \pi \]
Thus,
\[ V_{\text{total}} = \frac{108}{3} \pi + \frac{256}{3} \pi + \frac{500}{3} \pi = \frac{108 + 256 + 500}{3} \pi = \frac{864}{3} \pi = 288 \pi \, \text{cm}^3 \]

4. **Set the Total Volume Equal to the Volume of the Big Ball**:
Let the radius of the big ball be \( R \). The volume of the big ball is:
\[ V_{\text{big}} = \frac{4}{3} \pi R^3 \]
Setting the total volume equal to the volume of the big ball gives:
\[ \frac{4}{3} \pi R^3 = 288 \pi \]

5. **Solve for \( R^3 \)**:
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} R^3 = 288 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ R^3 = 288 \times \frac{3}{4} = 216 \]

6. **Find \( R \)**:
Taking the cube root of both sides:
\[ R = \sqrt[3]{216} = 6 \, \text{cm} \]

### Final Answer:
The radius of the big ball is \( R = 6 \) cm.