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

- For the first sphere (radius = 3 cm):
\[ V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi \, \text{cm}^3 \]

- For the second sphere (radius = 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 sphere (radius = 5 cm):
\[ V_3 = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3}\pi \, \text{cm}^3 \]

2. **Sum the Volumes of the Smaller Spheres**:
The total volume \( V_{total} \) of the three spheres is:
\[ V_{total} = V_1 + V_2 + V_3 = 36\pi + \frac{256}{3}\pi + \frac{500}{3}\pi \]
To add these, convert \( 36\pi \) into a fraction:
\[ 36\pi = \frac{108}{3}\pi \]
Now, combine the volumes:
\[ V_{total} = \frac{108}{3}\pi + \frac{256}{3}\pi + \frac{500}{3}\pi = \frac{108 + 256 + 500}{3}\pi = \frac{864}{3}\pi = 288\pi \, \text{cm}^3 \]

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

4. **Solve for \( R^3 \)**:
Cancel \( \pi \) from both sides:
\[ 288 = \frac{4}{3} R^3 \]
Multiply both sides by \( \frac{3}{4} \):
\[ R^3 = 288 \times \frac{3}{4} = 216 \]

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

### Final Answer:
The radius of the big spherical ball is \( 6 \, \text{cm} \).