Three solid metallic balls of radii 3 cm, 4 cm and 5 cm are melted and moulded into a single solid ball. The raidus of the new ball is :

Options

A. 2 cm
B. 3 cm
C. 4 cm
D. 6 cm

Answer

To find the radius of the new solid ball formed by melting three smaller metallic 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 Ball**:
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 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:
\[ V_3 = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3}\pi \text{ cm}^3 \]

2. **Calculate the Total Volume of the Smaller Balls**:
Now, we add the volumes of the three balls:
\[ V_{total} = V_1 + V_2 + V_3 = 36\pi + \frac{256}{3}\pi + \frac{500}{3}\pi \]

To add these volumes, we convert \( 36\pi \) into a fraction:
\[ 36\pi = \frac{108}{3}\pi \]
Now, we can sum them:
\[ V_{total} = \frac{108}{3}\pi + \frac{256}{3}\pi + \frac{500}{3}\pi = \frac{864}{3}\pi \text{ cm}^3 \]

3. **Set the Total Volume Equal to the Volume of the New Ball**:
Let the radius of the new ball be \( R \). The volume of the new ball is:
\[ V_{new} = \frac{4}{3} \pi R^3 \]
Setting the total volume equal to the volume of the new ball:
\[ \frac{4}{3} \pi R^3 = \frac{864}{3}\pi \]

4. **Cancel \( \frac{4}{3} \pi \)**:
We can cancel \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = 216 \]

5. **Calculate the Radius \( R \)**:
To find \( R \), we take the cube root of both sides:
\[ R = \sqrt[3]{216} = 6 \text{ cm} \]

### Final Answer:
The radius of the new solid ball is **6 cm**.

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